Old page wikitext, before the edit ($1) (old_wikitext) | '{{short description|Polynomial function of degree four}}{{Distinguish|Quantic (disambiguation){{!}}Quantic}}{{about|the univariate case|the bivariate case|Quartic plane curve}}{{redirect|Biquadratic function|the use in computer science|Biquadratic rational function}}{{Use dmy dates|date=December 2017}}[[File:Polynomialdeg4.svg|thumb|right|233px|Graph of a polynomial of degree 4, with 3 [[critical point (mathematics)|critical points]] and four [[real number|real]] [[root of a polynomial|roots]] (crossings of the ''x'' axis) (and thus no [[complex number|complex]] roots). If one or the other of the local [[minimum|minima]] were above the ''x'' axis, or if the local [[maximum]] were below it, or if there were no local maximum and one minimum below the ''x'' axis, there would only be two real roots (and two complex roots). If all three local extrema were above the ''x'' axis, or if there were no local maximum and one minimum above the ''x'' axis, there would be no real root (and four complex roots). The same reasoning applies in reverse to polynomial with a negative quartic coefficient.]]In [[algebra]], a '''quartic function''' is a [[function (mathematics)|function]] of the form:<math>f(x)=ax^4+bx^3+cx^2+dx+e,</math>{{ref|Alpha|α}}where ''a'' is nonzero,which is defined by a [[polynomial]] of [[Degree of a polynomial|degree]] four, called a '''quartic polynomial'''.A ''[[quartic equation]]'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form:<math>ax^4+bx^3+cx^2+dx+e=0 ,</math>where {{nowrap|''a'' ≠ 0}}.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Quartic Equation|url=https://mathworld.wolfram.com/QuarticEquation.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>The [[derivative]] of a quartic function is a [[cubic function]].Sometimes the term '''biquadratic''' is used instead of ''quartic'', but, usually, '''biquadratic function''' refers to a [[quadratic function]] of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form:<math>f(x)=ax^4+cx^2+e.</math>Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative [[infinity]]. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a [[Maxima and minima|global minimum]]. Likewise, if ''a'' is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.The degree four (''quartic'' case) is the highest degree such that every polynomial equation can be solved by [[Nth root|radicals]], according to the [[Abel–Ruffini theorem]].==History==[[Lodovico Ferrari]] is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a [[cubic equation|cubic]] to be found, it could not be published immediately.<ref>{{MacTutor|id=Ferrari|title=Lodovico Ferrari}}</ref> The solution of the quartic was published together with that of the cubic by Ferrari's mentor [[Gerolamo Cardano]] in the book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]''.<ref>{{Citation | last = Cardano | first = Gerolamo | author-link = Gerolamo Cardano | year = 1993 | orig-year = 1545 | title = Ars magna or The Rules of Algebra | publisher = Dover | isbn = 0-486-67811-3 | url-access = registration | url = https://archive.org/details/arsmagnaorruleso0000card }}</ref>The Soviet historian I. Y. Depman <small>([[:ru:Депман, Иван Яковлевич|ru]])</small> claimed that even earlier, in 1486, Spanish mathematician Valmes was [[burned at the stake]] for claiming to have solved the quartic equation.<ref>{{citation|last=Depman|title=Rasskazy o matematike|publisher=Gosdetizdat|year=1954|place=Leningrad|language=ru}}</ref> [[Inquisitor General]] [[Tomás de Torquemada]] allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.<ref>{{cite book |author=P. Beckmann |title=A history of π |publisher=Macmillan |year=1971 |page=80 |isbn=9780312381851 |url=https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80}}</ref> However, [[Petr Beckmann]], who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda.<ref>{{cite book |author=P. Beckmann |title=A history of π |publisher=Macmillan |year=1971 |page=191 |isbn=9780312381851 |url=https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80}}</ref> Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed.<ref>{{cite journal|author=P. Zoll | title=Letter to the Editor |journal=American Mathematical Monthly |volume=96 |issue=8 |year=1989 |pages=709–710 |jstor=2324719}}</ref>The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the [[Abel–Ruffini theorem]] in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by [[Évariste Galois]] prior to dying in a duel in 1832 later led to an elegant [[Galois theory|complete theory]] of the roots of polynomials, of which this theorem was one result.<ref>Stewart, Ian, ''Galois Theory, Third Edition'' (Chapman & Hall/CRC Mathematics, 2004)</ref>==Applications==Each [[coordinate]] of the intersection points of two [[conic section]]s is a solution of a quartic equation. The same is true for the intersection of a line and a [[torus]]. It follows that quartic equations often arise in [[computational geometry]] and all related fields such as [[computer graphics]], [[computer-aided design]], [[computer-aided manufacturing]] and [[optics]]. Here are examples of other geometric problems whose solution involves solving a quartic equation.In [[computer-aided manufacturing]], the torus is a shape that is commonly associated with the [[endmill]] cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the {{math|''z''}}-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.<ref>{{Cite web|url=http://people.math.gatech.edu/~etnyre/class/4441Fall16/ShifrinDiffGeo.pdf|title=DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces, p. 36|website=math.gatech.edu}}</ref>A quartic equation arises also in the process of solving the [[crossed ladders problem]], in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Crossed Ladders Problem|url=https://mathworld.wolfram.com/CrossedLaddersProblem.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>In optics, [[Alhazen's problem]] is "''Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.''" This leads to a quartic equation.<ref name=MacTutor>{{MacTutor|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref><ref>{{citation|title=Scientific Method, Statistical Method and the Speed of Light|first1=R. J.|last1=MacKay|first2=R. W.|last2=Oldford|journal=Statistical Science|volume=15|issue=3|date=August 2000|pages=254–78|doi=10.1214/ss/1009212817|mr=1847825|doi-access=free}}</ref><ref name=Weiss>{{Citation|last = Neumann|first = Peter M.|author-link = Peter M. Neumann|journal = [[American Mathematical Monthly]]|title = Reflections on Reflection in a Spherical Mirror|year = 1998|volume = 105|issue = 6|pages = 523–528|doi = 10.2307/2589403|jstor = 2589403}}</ref>Finding the [[distance of closest approach of ellipses and ellipsoids#Distance of closest approach of two ellipses|distance of closest approach of two ellipses]] involves solving a quartic equation.The [[eigenvalue]]s of a 4×4 [[matrix (mathematics)|matrix]] are the roots of a quartic polynomial which is the [[characteristic polynomial]] of the matrix.The characteristic equation of a fourth-order linear [[difference equation]] or [[differential equation]] is a quartic equation. An example arises in the [[Bending#Timoshenko-Rayleigh theory|Timoshenko-Rayleigh theory]] of beam bending.<ref>{{Cite book|last=Shabana|first=A. A.|url=https://books.google.com/books?id=G2UyBTji18oC&q=Timoshenko-Rayleigh+theory&pg=PA2|title=Theory of Vibration: An Introduction|date=1995-12-08|publisher=Springer Science & Business Media|isbn=978-0-387-94524-8|language=en}}</ref>[[Intersection (Euclidean geometry)|Intersections]] between spheres, cylinders, or other [[quadric]]s can be found using quartic equations.==Inflection points and golden ratio==Letting {{mvar|F}} and {{mvar|G}} be the distinct [[inflection point]]s of the graph of a quartic function, and letting {{mvar|H}} be the intersection of the inflection [[secant line]] {{mvar|FG}} and the quartic, nearer to {{mvar|G}} than to {{mvar|F}}, then {{mvar|G}} divides {{mvar|FH}} into the [[golden section]]:<ref>{{Citation|last = Aude|first = H. T. R.|journal = [[American Mathematical Monthly]]|year = 1949|issue = 3|volume = 56|title = Notes on Quartic Curves|jstor = 2305030|doi = 10.2307/2305030|pages=165–170}}</ref>:<math>\frac{FG}{GH}=\frac{1+\sqrt{5}}{2}= \varphi \; (\text{the golden ratio}).</math>Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.==Solution=====Nature of the roots===Given the general quartic equation:<math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>with real coefficients and {{math|''a'' ≠ 0}} the nature of its roots is mainly determined by the sign of its [[discriminant]] :<math>\begin{align} \Delta = {} &256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 \\ &+ 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e \\&- 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2\end{align} </math> This may be refined by considering the signs of four other polynomials::<math>P = 8ac - 3b^2</math>such that {{math|{{sfrac|''P''|8''a''<sup>2</sup>}}}} is the second degree coefficient of the associated depressed quartic (see [[#Converting_to_a_depressed_quartic|below]]);:<math>R= b^3+8da^2-4abc,</math>such that {{math|{{sfrac|''R''|8''a''<sup>3</sup>}}}} is the first degree coefficient of the associated depressed quartic; :<math>\Delta_0 = c^2 - 3bd + 12ae,</math>which is 0 if the quartic has a triple root; and:<math>D = 64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 bd - 3 b^4</math>which is 0 if the quartic has two double roots.The possible cases for the nature of the roots are as follows:<ref>{{cite journal|first= E. L.|last=Rees|title=Graphical Discussion of the Roots of a Quartic Equation|journal = The American Mathematical Monthly|volume=29|issue=2|year=1922|pages=51–55|doi=10.2307/2972804|jstor = 2972804}}</ref>* If {{math|∆ < 0}} then the equation has two distinct real roots and two [[complex conjugate]] non-real roots.* If {{math|∆ > 0}} then either the equation's four roots are all real or none is.** If {{mvar|P}} < 0 and {{mvar|D}} < 0 then all four roots are real and distinct.** If {{mvar|P}} > 0 or {{mvar|D}} > 0 then there are two pairs of non-real complex conjugate roots.<ref>{{Cite journal | last1 = Lazard | first1 = D. | doi = 10.1016/S0747-7171(88)80015-4 | title = Quantifier elimination: Optimal solution for two classical examples | journal = Journal of Symbolic Computation | volume = 5 | pages = 261–266 | year = 1988 | issue = 1–2 | doi-access = free }}</ref>* If {{math|∆ {{=}} 0}} then (and only then) the polynomial has a [[multiplicity (mathematics)|multiple]] root. Here are the different cases that can occur:** If {{mvar|P}} < 0 and {{mvar|D}} < 0 and {{math|∆<sub>0</sub> ≠ 0}}, there are a real double root and two real simple roots.** If {{mvar|D}} > 0 or ({{mvar|P}} > 0 and ({{mvar|D}} ≠ 0 or {{mvar|R}} ≠ 0)), there are a real double root and two complex conjugate roots.** If {{math|∆<sub>0</sub> {{=}} 0}} and {{mvar|D}} ≠ 0, there are a triple root and a simple root, all real.** If {{mvar|D}} = 0, then:***If {{mvar|P}} < 0, there are two real double roots.***If {{mvar|P}} > 0 and {{mvar|R}} = 0, there are two complex conjugate double roots.***If {{math|∆<sub>0</sub> {{=}} 0}}, all four roots are equal to {{math|−{{sfrac|''b''|4''a''}}}}There are some cases that do not seem to be covered, but in fact they cannot occur. For example, {{math|∆<sub>0</sub> > 0}}, {{mvar|P}} = 0 and {{mvar|D}} ≤ 0 is not one of the cases. In fact, if {{math|∆<sub>0</sub> > 0}} and {{mvar|P}} = 0 then {{mvar|D}} > 0, since <math>16 a^2\Delta_0 = 3D + P^2; </math> so this combination is not possible.===General formula for roots===[[File:Quartic Formula.svg|thumb|600px|right|Solution of <math>x^4+ax^3+bx^2+cx+d=0</math> written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.]]The four roots {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''x''<sub>3</sub>}}, and {{math|''x''<sub>4</sub>}} for the general quartic equation:<math>ax^4+bx^3+cx^2+dx+e=0 \,</math>with {{mvar|a}} ≠ 0 are given in the following formula, which is deduced from the one in the section on [[#Ferrari's solution|Ferrari's method]] by back changing the variables (see {{slink||Converting to a depressed quartic}}) and using the formulas for the [[Quadratic function|quadratic]] and [[Cubic function#General formula for roots|cubic equation]]s.:<math>\begin{align}x_{1,2}\ &= -\frac{b}{4a} - S \pm \frac12\sqrt{-4S^2 - 2p + \frac{q}{S}}\\x_{3,4}\ &= -\frac{b}{4a} + S \pm \frac12\sqrt{-4S^2 - 2p - \frac{q}{S}}\end{align}</math>where {{mvar|p}} and {{mvar|q}} are the coefficients of the second and of the first degree respectively in the [[#Converting to a depressed quartic|associated depressed quartic]]:<math>\begin{align}p &= \frac{8ac-3b^2}{8a^2}\\q &= \frac{b^3 - 4abc + 8a^2d}{8a^3} \end{align}</math>:and where:<math>\begin{align}S &= \frac{1}{2}\sqrt{-\frac23\ p+\frac{1}{3a}\left(Q + \frac{\Delta_0}{Q}\right)}\\Q &= \sqrt[3]{\frac{\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2}} \end{align}</math>(if {{math|''S'' {{=}} 0}} or {{math|''Q'' {{=}} 0}}, see {{slink||Special cases of the formula}}, below)with:<math>\begin{align}\Delta_0 &= c^2 - 3bd + 12ae\\\Delta_1 &= 2c^3 - 9bcd + 27b^2 e + 27ad^2 - 72ace\end{align}</math>and:<math>\Delta_1^2-4\Delta_0^3 = - 27 \Delta\ ,</math> where <math>\Delta</math> is the aforementioned [[discriminant]]. For the cube root expression for ''Q'', any of the three cube roots in the complex plane can be used, although if one of them is real that is the natural and simplest one to choose. The mathematical expressions of these last four terms are very similar to those of their [[Cubic function#Algebraic solution|cubic counterparts]].====Special cases of the formula====*If <math>\Delta > 0,</math> the value of <math>Q</math> is a non-real complex number. In this case, either all roots are non-real or they are all real. In the latter case, the value of <math>S</math> is also real, despite being expressed in terms of <math>Q;</math> this is [[casus irreducibilis]] of the cubic function extended to the present context of the quartic. One may prefer to express it in a purely real way, by using [[trigonometric functions]], as follows:::<math>S = \frac{1}{2} \sqrt{-\frac23\ p+\frac{2}{3a}\sqrt{\Delta_0}\cos\frac{\varphi}{3}}</math>:where::<math>\varphi = \arccos\left(\frac{\Delta_1}{2\sqrt{\Delta_0^3}}\right).</math>*If <math>\Delta \neq 0</math> and <math>\Delta_0 = 0,</math> the sign of <math>\sqrt{\Delta_1^2 - 4 \Delta_0^3}=\sqrt{\Delta_1^2} </math> has to be chosen to have <math>Q \neq 0,</math> that is one should define <math>\sqrt{\Delta_1^2}</math> as <math>\Delta_1,</math> maintaining the sign of <math>\Delta_1.</math>*If <math>S = 0,</math> then one must change the choice of the cube root in <math>Q</math> in order to have <math>S \neq 0.</math> This is always possible except if the quartic may be factored into <math>\left(x+\tfrac{b}{4a}\right)^4.</math> The result is then correct, but misleading because it hides the fact that no cube root is needed in this case. In fact this case{{Clarify|reason=Both kinds or specific?|date=March 2024}} may occur only if the [[numerator]] of <math>q</math> is zero, in which case the associated [[#Converting to a depressed quartic|depressed quartic]] is biquadratic; it may thus be solved by the method described [[#Biquadratic equation|below]].*If <math>\Delta = 0</math> and <math>\Delta_0 = 0,</math> and thus also <math>\Delta_1 = 0,</math> at least three roots are equal to each other, and the roots are [[rational function]]s of the coefficients. The triple root <math>x_0</math> is a common root of the quartic and its second derivative <math>2(6ax^2+3bx+c);</math> it is thus also the unique root of the remainder of the [[Euclidean division]] of the quartic by its second derivative, which is a linear polynomial. The simple root <math>x_1</math> can be deduced from <math>x_1+3x_0=-b/a.</math>*If <math>\Delta=0</math> and <math> \Delta_0 \neq 0,</math> the above expression for the roots is correct but misleading, hiding the fact that the polynomial is [[irreducible polynomial|reducible]] and no cube root is needed to represent the roots.===Simpler cases=======Reducible quartics====Consider the general quartic:<math>Q(x) = a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.</math>It is [[irreducible polynomial|reducible]] if {{math|''Q''(''x'') {{=}} ''R''(''x'')×''S''(''x'')}}, where {{math|''R''(''x'')}} and {{math|''S''(''x'')}} are non-constant polynomials with [[rational number|rational]] coefficients (or more generally with coefficients in the same [[field (mathematics)|field]] as the coefficients of {{math|''Q''(''x'')}}). Such a factorization will take one of two forms::<math>Q(x) = (x-x_1)(b_3x^3+b_2x^2+b_1x+b_0)</math>or:<math>Q(x) = (c_2x^2+c_1x+c_0)(d_2x^2+d_1x+d_0).</math>In either case, the roots of {{math|''Q''(''x'')}} are the roots of the factors, which may be computed using the formulas for the roots of a [[quadratic function]] or [[cubic function]].Detecting the existence of such factorizations can be done [[Resolvent cubic#Factoring quartic polynomials|using the resolvent cubic of {{math|''Q''(''x'')}}]]. It turns out that:* if we are working over {{math|'''R'''}} (that is, if coefficients are restricted to be real numbers) (or, more generally, over some [[real closed field]]) then there is always such a factorization;* if we are working over {{math|'''Q'''}} (that is, if coefficients are restricted to be rational numbers) then there is an algorithm to determine whether or not {{math|''Q''(''x'')}} is reducible and, if it is, how to express it as a product of polynomials of smaller degree.In fact, several methods of solving quartic equations ([[Quartic function#Ferrari's solution|Ferrari's method]], [[Quartic function#Descartes' solution|Descartes' method]], and, to a lesser extent, [[Quartic function#Euler's solution|Euler's method]]) are based upon finding such factorizations.====Biquadratic equation====If {{math|''a''<sub>3</sub> {{=}} ''a''<sub>1</sub> {{=}} 0}} then the function :<math>Q(x) = a_4x^4+a_2x^2+a_0</math> is called a '''biquadratic function'''; equating it to zero defines a '''biquadratic equation''', which is easy to solve as followsLet the auxiliary variable {{math|''z'' {{=}} ''x''<sup>2</sup>}}.Then {{math|''Q''(''x'')}} becomes a [[Quadratic function|quadratic]] {{math|''q''}} in {{math|''z''}}: {{math|''q''(''z'') {{=}} ''a''<sub>4</sub>''z''<sup>2</sup> + ''a''<sub>2</sub>''z'' + ''a''<sub>0</sub>}}. Let {{math|''z''<sub>+</sub>}} and {{math|''z''<sub>−</sub>}} be the roots of {{math|''q''(''z'')}}. Then the roots of the quartic {{math|''Q''(''x'')}} are:<math>\begin{align}x_1&=+\sqrt{z_+},\\x_2&=-\sqrt{z_+},\\x_3&=+\sqrt{z_-},\\x_4&=-\sqrt{z_-}.\end{align}</math>==== Quasi-palindromic equation ====The polynomial: <math>P(x)=a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2</math>is almost [[Reciprocal polynomial#Palindromic polynomial|palindromic]], as {{math|''P''(''mx'') {{=}} {{sfrac|''x''<sup>4</sup>|''m''<sup>2</sup>}}''P''({{sfrac|''m''|''x''}})}} (it is palindromic if {{math|''m'' {{=}} 1}}). The change of variables {{math|''z'' {{=}} ''x'' + {{sfrac|''m''|''x''}}}} in {{math|{{sfrac|''P''(''x'')|''x''<sup>2</sup>}} {{=}} 0}} produces the [[quadratic equation]] {{math|''a''<sub>0</sub>''z''<sup>2</sup> + ''a''<sub>1</sub>''z'' + ''a''<sub>2</sub> − 2''ma''<sub>0</sub> {{=}} 0}}. Since {{math|''x''<sup>2</sup> − ''xz'' + ''m'' {{=}} 0}}, the quartic equation {{math|''P''(''x'') {{=}} 0}} may be solved by applying the [[quadratic formula]] twice.===Solution methods=======Converting to a depressed quartic====For solving purposes, it is generally better to convert the quartic into a '''depressed quartic''' by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.Let:<math> a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 </math>be the general quartic equation we want to solve.Dividing by {{math|''a''<sub>4</sub>}}, provides the equivalent equation {{math|''x''<sup>4</sup> + ''bx''<sup>3</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e'' {{=}} 0}}, with {{math|''b'' {{=}} {{sfrac|''a''<sub>3</sub>|''a''<sub>4</sub>}}}}, {{math|''c'' {{=}} {{sfrac|''a''<sub>2</sub>|''a''<sub>4</sub>}}}}, {{math|''d'' {{=}} {{sfrac|''a''<sub>1</sub>|''a''<sub>4</sub>}}}}, and {{math|''e'' {{=}} {{sfrac|''a''<sub>0</sub>|''a''<sub>4</sub>}}}}.Substituting {{math|''y'' − {{sfrac|''b''|4}}}} for {{mvar|x}} gives, after regrouping the terms, the equation {{math|''y''<sup>4</sup> + ''py''<sup>2</sup> + ''qy'' + ''r'' {{=}} 0}},where:<math>\begin{align}p&=\frac{8c-3b^2}{8} =\frac{8a_2a_4-3{a_3}^2}{8{a_4}^2}\\q&=\frac{b^3-4bc+8d}{8} =\frac{{a_3}^3-4a_2a_3a_4+8a_1{a_4}^2}{8{a_4}^3}\\r&=\frac{-3b^4+256e-64bd+16b^2c}{256}=\frac{-3{a_3}^4+256a_0{a_4}^3-64a_1a_3{a_4}^2+16a_2{a_3}^2a_4}{256{a_4}^4}.\end{align}</math>If {{math|''y''<sub>0</sub>}} is a root of this depressed quartic, then {{math|''y''<sub>0</sub> − {{sfrac|''b''|4}}}} (that is {{math|''y''<sub>0</sub> − {{sfrac|''a''<sub>3</sub>|4''a''<sub>4</sub>}})}} is a root of the original quartic and every root of the original quartic can be obtained by this process.====Ferrari's solution====As explained in the preceding section, we may start with the ''depressed quartic equation'':<math> y^4 + p y^2 + q y + r = 0. </math>This depressed quartic can be solved by means of a method discovered by [[Lodovico Ferrari]]. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as:<math> \left(y^2 + \frac p2\right)^2 = -q y - r + \frac{p^2}4. </math>Then, we introduce a variable {{mvar|m}} into the factor on the left-hand side by adding {{math|2''y''<sup>2</sup>''m'' + ''pm'' + ''m''<sup>2</sup>}} to both sides. After regrouping the coefficients of the power of {{mvar|y}} on the right-hand side, this gives the equation {{NumBlk|:|<math> \left(y^2 + \frac p2 + m\right)^2 = 2 m y^2 - q y + m^2 + m p + \frac{p^2}4 - r, </math>|{{EquationRef|1}}}}which is equivalent to the original equation, whichever value is given to {{mvar|m}}.As the value of {{mvar|m}} may be arbitrarily chosen, we will choose it in order to [[complete the square]] on the right-hand side. This implies that the [[discriminant]] in {{mvar|y}} of this [[quadratic equation]] is zero, that is {{mvar|m}} is a root of the equation:<math> (-q)^2 - 4 (2m)\left(m^2 + p m + \frac{p^2}4 - r\right) = 0,\,</math>which may be rewritten as{{NumBlk|:|<math>8m^3+ 8pm^2 + (2p^2 -8r)m- q^2 =0.</math>|{{EquationRef|1a}}}}This is the [[resolvent cubic]] of the quartic equation. The value of {{mvar|m}} may thus be obtained from [[Cubic equation#Cardano's method|Cardano's formula]]. When {{mvar|m}} is a root of this equation, the right-hand side of equation (''{{EquationNote|1}}'') is the square:<math>\left(\sqrt{2m}y-\frac q{2\sqrt{2m}}\right)^2.</math>However, this induces a division by zero if {{math|''m'' {{=}} 0}}. This implies {{math|''q'' {{=}} 0}}, and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that {{math|''m'' ≠ 0}}. This is always possible except for the depressed equation {{math|''y''<sup>4</sup> {{=}} 0}}.Now, if {{mvar|m}} is a root of the cubic equation such that {{math|''m'' ≠ 0}}, equation (''{{EquationNote|1}}'') becomes:<math> \left(y^2 + \frac p2 + m\right)^2 = \left(y\sqrt{2 m}-\frac{q}{2\sqrt{2 m}}\right)^2. </math>This equation is of the form {{math|''M''<sup>2</sup> {{=}} ''N''<sup>2</sup>}}, which can be rearranged as {{math|''M''<sup>2</sup> − ''N''<sup>2</sup> {{=}} 0}} or {{math|(''M'' + ''N'')(''M'' − ''N'') {{=}} 0}}. Therefore, equation (''{{EquationNote|1}}'') may be rewritten as:<math> \left(y^2 + \frac p2 + m + \sqrt{2 m}y-\frac q{2\sqrt{2 m}}\right) \left(y^2 + \frac p2 + m - \sqrt{2 m}y+\frac q{2\sqrt{2 m}}\right)=0.</math>This equation is easily solved by applying to each factor the [[quadratic formula]]. Solving them we may write the four roots as:<math>y={\pm_1\sqrt{2 m} \pm_2 \sqrt{-\left(2p + 2m \pm_1 {\sqrt 2q \over \sqrt{m}} \right)} \over 2},</math>where {{math|±<sub>1</sub>}} and {{math|±<sub>2</sub>}} denote either {{math|+}} or {{math|−}}. As the two occurrences of {{math|±<sub>1</sub>}} must denote the same sign, this leaves four possibilities, one for each root.Therefore, the solutions of the original quartic equation are:<math>x=-{a_3 \over 4a_4} + {\pm_1\sqrt{2 m} \pm_2 \sqrt{-\left(2p + 2m \pm_1 {\sqrt2q \over \sqrt{m}} \right)} \over 2}.</math> A comparison with the [[#General_formula_for_roots|general formula]] above shows that {{math|{{sqrt|2''m''}} {{=}} 2''S''}}.====Descartes' solution====Descartes<ref>{{Citation|last = Descartes|first = René|author-link = René Descartes|title = [[La Géométrie|The Geometry of Rene Descartes with a facsimile of the first edition]]|isbn = 0-486-60068-8|publisher = [[Dover Publications|Dover]]|year = 1954|jfm = 51.0020.07|chapter = Book III: On the construction of solid and supersolid problems|orig-year = 1637}}</ref> introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let:<math> \begin{align} x^4 + bx^3 + cx^2 + dx + e & = (x^2 + sx + t)(x^2 + ux + v) \\ & = x^4 + (s + u)x^3 + (t + v + su)x^2 + (sv + tu)x + tv \end{align} </math>By [[equating coefficients]], this results in the following system of equations::<math> \left\{\begin{array}{l} b = s + u \\ c = t + v + su \\ d = sv + tu \\ e = tv \end{array}\right. </math>This can be simplified by starting again with the [[#Converting to a depressed quartic|depressed quartic]] {{math|''y''<sup>4</sup> + ''py''<sup>2</sup> + ''qy'' + ''r''}}, which can be obtained by substituting {{math|''y'' − ''b''/4}} for {{math|''x''}}. Since the coefficient of {{math|''y''<sup>3</sup>}} is {{math|0}}, we get {{math|''s'' {{=}} −''u''}}, and::<math> \left\{\begin{array}{l} p + u^2 = t + v \\ q = u (t - v) \\ r = tv \end{array}\right. </math>One can now eliminate both {{mvar|t}} and {{mvar|v}} by doing the following::<math> \begin{align} u^2(p + u^2)^2 - q^2 & = u^2(t + v)^2 - u^2(t - v)^2 \\ & = u^2 [(t + v + (t - v))(t + v - (t - v))]\\ & = u^2(2t)(2v) \\ & = 4u^2tv \\ & = 4u^2r \end{align} </math>If we set {{math|''U'' {{=}} ''u''<sup>2</sup>}}, then solving this equation becomes finding the roots of the [[resolvent cubic]]{{NumBlk|:|<math> U^3 + 2pU^2 + (p^2-4r)U - q^2,</math>|{{EquationRef|2}}}}which is [[Cubic function#General solution to the cubic equation with real coefficients|done elsewhere]]. This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m.If {{math|''u''}} is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic {{math|''x''<sup>4</sup>}}, which is trivially factored),:<math> \left\{\begin{array}{l} s = -u \\ 2t = p + u^2 + q/u \\ 2v = p + u^2 - q/u \end{array}\right. </math>The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of {{mvar|u}} for the square root of {{mvar|U}} merely exchanges the two quadratics with one another.The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic (''{{EquationNote|2}}'') has a non-zero root which is the square of a rational, or {{math|''p''<sup>2</sup> − 4''r''}} is the square of rational and {{math|''q'' {{=}} 0}}; this can readily be checked using the [[rational root test]].<ref name=Brookfield>{{cite journal |author=Brookfield, G. |title=Factoring quartic polynomials: A lost art |journal=[[Mathematics Magazine]] |volume=80 |issue=1 |year=2007 |pages=67–70|doi=10.1080/0025570X.2007.11953453 |s2cid=53375377 |url = https://www.maa.org/sites/default/files/Brookfield2007-103574.pdf}}</ref>====Euler's solution====A variant of the previous method is due to [[Leonhard Euler|Euler]].<ref>{{Citation|last = van der Waerden|first=Bartel Leendert|author-link = Bartel Leendert van der Waerden|title = [[Moderne Algebra|Algebra]]|volume = 1|publisher=[[Springer Science+Business Media|Springer-Verlag]]|edition = 7th|isbn = 0-387-97424-5|year = 1991|section = The Galois theory: Equations of the second, third, and fourth degrees|zbl = 0724.12001}}</ref><ref>{{Citation|last = Euler|first = Leonhard|author-link = Leonhard Euler|title = [[Elements of Algebra]]|chapter= Of a new method of resolving equations of the fourth degree|publisher=[[Springer Science+Business Media|Springer-Verlag]]|orig-year = 1765|year = 1984|zbl = 0557.01014|isbn = 978-1-4613-8511-0}}</ref> Unlike the previous methods, both of which use ''some'' root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r''}}. Observe that, if* {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r'' {{=}} (''x''<sup>2</sup> + ''sx'' + ''t'')(''x''<sup>2</sup> − ''sx'' + ''v'')}},* {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are the roots of {{math|''x''<sup>2</sup> + ''sx'' + ''t''}},* {{math|''r''<sub>3</sub>}} and {{math|''r''<sub>4</sub>}} are the roots of {{math|''x''<sup>2</sup> − ''sx'' + ''v''}},then* the roots of {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r''}} are {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}},* {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> {{=}} −''s''}},* {{math|''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} ''s''}}.Therefore, {{math|(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>) {{=}} −''s''<sup>2</sup>}}. In other words, {{math|−(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>)}} is one of the roots of the resolvent cubic (''{{EquationNote|2}}'') and this suggests that the roots of that cubic are equal to {{math|−(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>)}}, {{math|−(''r''<sub>1</sub> + ''r''<sub>3</sub>)(''r''<sub>2</sub> + ''r''<sub>4</sub>)}}, and {{math|−(''r''<sub>1</sub> + ''r''<sub>4</sub>)(''r''<sub>2</sub> + ''r''<sub>3</sub>)}}. This is indeed true and it follows from [[Vieta's formulas]]. It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> + ''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} 0}}. (Of course, this also follows from the fact that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> + ''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} −''s'' + ''s''}}.) Therefore, if {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are the roots of the resolvent cubic, then the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} are such that:<math>\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\(r_1+r_2)(r_3+r_4)=-\alpha\\(r_1+r_3)(r_2+r_4)=-\beta\\(r_1+r_4)(r_2+r_3)=-\gamma\text{.}\end{array}\right.</math>It is a consequence of the first two equations that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub>}} is a square root of {{math|''α''}} and that {{math|''r''<sub>3</sub> + ''r''<sub>4</sub>}} is the other square root of {{math|''α''}}. For the same reason,* {{math|''r''<sub>1</sub> + ''r''<sub>3</sub>}} is a square root of {{math|''β''}},* {{math|''r''<sub>2</sub> + ''r''<sub>4</sub>}} is the other square root of {{math|''β''}},* {{math|''r''<sub>1</sub> + ''r''<sub>4</sub>}} is a square root of {{math|''γ''}},* {{math|''r''<sub>2</sub> + ''r''<sub>3</sub>}} is the other square root of {{math|''γ''}}.Therefore, the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} are such that:<math>\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\r_1+r_2=\sqrt{\alpha}\\r_1+r_3=\sqrt{\beta}\\r_1+r_4=\sqrt{\gamma}\text{;}\end{array}\right.</math>the sign of the square roots will be dealt with below. The only solution of this system is::<math>\left\{\begin{array}{l}r_1=\frac{\sqrt{\alpha}+\sqrt{\beta}+\sqrt{\gamma}}2\\[2mm]r_2=\frac{\sqrt{\alpha}-\sqrt{\beta}-\sqrt{\gamma}}2\\[2mm]r_3=\frac{-\sqrt{\alpha}+\sqrt{\beta}-\sqrt{\gamma}}2\\[2mm]r_4=\frac{-\sqrt{\alpha}-\sqrt{\beta}+\sqrt{\gamma}}2\text{.}\end{array}\right.</math>Since, in general, there are two choices for each square root, it might look as if this provides {{math|8 ({{=}} 2<sup>3</sup>)}} choices for the set {{math|{''r''<sub>1</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, ''r''<sub>4</sub>}}}, but, in fact, it provides no more than {{math|2}} such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set {{math|{''r''<sub>1</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, ''r''<sub>4</sub>}}} becomes the set {{math|{−''r''<sub>1</sub>, −''r''<sub>2</sub>, −''r''<sub>3</sub>, −''r''<sub>4</sub>}}}.In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} and uses them to compute the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} from the previous equalities. Then, one computes the number {{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}}}}. Since {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are the roots of (''{{EquationNote|2}}''), it is a consequence of Vieta's formulas that their product is equal to {{math|''q''<sup>2</sup>}} and therefore that {{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}} {{=}} ±''q''}}. But a straightforward computation shows that:{{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}} {{=}} ''r''<sub>1</sub>''r''<sub>2</sub>''r''<sub>3</sub> + ''r''<sub>1</sub>''r''<sub>2</sub>''r''<sub>4</sub> + ''r''<sub>1</sub>''r''<sub>3</sub>''r''<sub>4</sub> + ''r''<sub>2</sub>''r''<sub>3</sub>''r''<sub>4</sub>.}}If this number is {{math|−''q''}}, then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be {{math|−''r''<sub>1</sub>}}, {{math|−''r''<sub>2</sub>}}, {{math|−''r''<sub>3</sub>}}, and {{math|−''r''<sub>4</sub>}}, which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one).This argument suggests another way of choosing the square roots:* pick ''any'' square root {{math|{{sqrt|''α''}}}} of {{math|''α''}} and ''any'' square root {{math|{{sqrt|''β''}}}} of {{math|''β''}};* ''define'' {{math|{{sqrt|''γ''}}}} as <math>-\frac q{\sqrt{\alpha}\sqrt{\beta}}</math>.Of course, this will make no sense if {{math|''α''}} or {{math|''β''}} is equal to {{math|0}}, but {{math|0}} is a root of (''{{EquationNote|2}}'') only when {{math|''q'' {{=}} 0}}, that is, only when we are dealing with a [[Quartic function#Biquadratic equation|biquadratic equation]], in which case there is a much simpler approach.====Solving by Lagrange resolvent====The [[symmetric group]] {{math|''S''<sub>4</sub>}} on four elements has the [[Klein four-group]] as a [[normal subgroup]]. This suggests using a '''{{visible anchor|resolvent cubic}}''' whose roots may be variously described as a discrete Fourier transform or a [[Hadamard matrix]] transform of the roots; see [[Lagrange resolvents]] for the general method. Denote by {{math|''x<sub>i</sub>''}}, for {{math|''i''}} from {{math|0}} to {{math|3}}, the four roots of {{math|''x''<sup>4</sup> + ''bx''<sup>3</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e''}}. If we set: <math> \begin{align}s_0 &= \tfrac12(x_0 + x_1 + x_2 + x_3), \\[4pt]s_1 &= \tfrac12(x_0 - x_1 + x_2 - x_3), \\[4pt]s_2 &= \tfrac12(x_0 + x_1 - x_2 - x_3), \\[4pt]s_3 &= \tfrac12(x_0 - x_1 - x_2 + x_3),\end{align}</math>then since the transformation is an [[Involution (mathematics)|involution]] we may express the roots in terms of the four {{math|''s<sub>i</sub>''}} in exactly the same way. Since we know the value {{math|''s''<sub>0</sub> {{=}} −{{sfrac|''b''|2}}}}, we only need the values for {{math|''s''<sub>1</sub>}}, {{math|''s''<sub>2</sub>}} and {{math|''s''<sub>3</sub>}}. These are the roots of the polynomial:<math>(s^2 - {s_1}^2)(s^2-{s_2}^2)(s^2-{s_3}^2).</math>Substituting the {{math|''s<sub>i</sub>''}} by their values in term of the {{math|''x<sub>i</sub>''}}, this polynomial may be expanded in a polynomial in {{math|''s''}} whose coefficients are [[symmetric polynomial]]s in the {{math|''x<sub>i</sub>''}}. By the [[fundamental theorem of symmetric polynomials]], these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is {{math|''b'' {{=}} 0}}, this results in the polynomial{{NumBlk|:|<math> s^6+2cs^4+(c^2-4e)s^2-d^2 </math>|{{EquationRef|3}}}}This polynomial is of degree six, but only of degree three in {{math|''s''<sup>2</sup>}}, and so the corresponding equation is solvable by the method described in the article about [[cubic function]]. By substituting the roots in the expression of the {{math|''x<sub>i</sub>''}} in terms of the {{math|''s<sub>i</sub>''}}, we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the {{math|''x<sub>i</sub>''}}.These expressions are unnecessarily complicated, involving the [[root of unity|cubic roots of unity]], which can be avoided as follows. If {{math|''s''}} is any non-zero root of (''{{EquationNote|3}}''), and if we set:<math>\begin{align}F_1(x) & = x^2 + sx + \frac{c}{2} + \frac{s^2}{2} - \frac{d}{2s} \\F_2(x) & = x^2 - sx + \frac{c}{2} + \frac{s^2}{2} + \frac{d}{2s}\end{align}</math>then:<math>F_1(x)\times F_2(x) = x^4 + cx^2 + dx + e.</math>We therefore can solve the quartic by solving for {{math|''s''}} and then solving for the roots of the two factors using the [[quadratic formula]].This gives exactly the same formula for the roots as the one provided by [[Quartic function#Descartes' solution|Descartes' method]].====Solving with algebraic geometry====There is an alternative solution using algebraic geometry<ref>{{Citation|last = Faucette|first = William M.|journal = [[American Mathematical Monthly]]|pages = 51–57|title = A Geometric Interpretation of the Solution of the General Quartic Polynomial|volume = 103|year = 1996|issue = 1|doi = 10.2307/2975214|jstor = 2975214|mr = 1369151}}</ref> In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three [[degenerate conic|reducible quadratic curves]] (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic.The four roots of the depressed quartic {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r'' {{=}} 0}} may also be expressed as the {{mvar|x}} coordinates of the intersections of the two quadratic equations {{math|''y''<sup>2</sup> + ''py'' + ''qx'' + ''r'' {{=}} 0}} and {{math|''y'' − ''x''<sup>2</sup> {{=}} 0}} i.e., using the substitution {{math|''y'' {{=}} ''x''<sup>2</sup>}} that two quadratics intersect in four points is an instance of [[Bézout's theorem]]. Explicitly, the four points are {{math|''P<sub>i</sub>'' ≔ (''x<sub>i</sub>'', ''x<sub>i</sub>''<sup>2</sup>)}} for the four roots {{math|''x<sub>i</sub>''}} of the quartic.These four points are not collinear because they lie on the irreducible quadratic {{math|''y'' {{=}} ''x''<sup>2</sup>}} and thus there is a 1-parameter family of quadratics (a [[pencil of curves]]) passing through these points. Writing the projectivization of the two quadratics as [[quadratic form]]s in three variables::<math>\begin{align}F_1(X,Y,Z) &:= Y^2 + pYZ + qXZ + rZ^2,\\F_2(X,Y,Z) &:= YZ - X^2\end{align}</math>the pencil is given by the forms {{math|''λF''<sub>1</sub> + ''μF''<sub>2</sub>}} for any point {{math|[''λ'', ''μ'']}} in the projective line — in other words, where {{math|''λ''}} and {{math|''μ''}} are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros.This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done <math>\textstyle{\binom{4}{2}}</math> = {{math|6}} different ways. Denote these {{math|''Q''<sub>1</sub> {{=}} ''L''<sub>12</sub> + ''L''<sub>34</sub>}}, {{math|''Q''<sub>2</sub> {{=}} ''L''<sub>13</sub> + ''L''<sub>24</sub>}}, and {{math|''Q''<sub>3</sub> {{=}} ''L''<sub>14</sub> + ''L''<sub>23</sub>}}. Given any two of these, their intersection has exactly the four points.The reducible quadratics, in turn, may be determined by expressing the quadratic form {{math|''λF''<sub>1</sub> + ''μF''<sub>2</sub>}} as a {{math|3×3}} matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in {{math|''λ''}} and {{math|''μ''}} and corresponds to the resolvent cubic.==See also==*{{annotated link|Linear function}}*{{annotated link|Quadratic function}}*{{annotated link|Cubic function}}*{{annotated link|Quintic function}}==Notes==:{{note|Alpha|α}} For the purposes of this article, ''e'' is used as a [[Variable (mathematics)|variable]] as opposed to its conventional use as [[e (mathematical constant)|Euler's number]](except when otherwise specified).==References=={{reflist}}==Further reading==* {{cite journal |author=Carpenter, W. |title=On the solution of the real quartic |journal=[[Mathematics Magazine]] |volume=39 |year=1966 |issue=1 |pages=28–30 |doi=10.2307/2688990|jstor=2688990 }}* {{cite journal |author1=Yacoub, M.D.|author2=Fraidenraich, G. |title=A solution to the quartic equation |journal=[[Mathematical Gazette]] |volume=96|date=July 2012|pages=271–275 |doi=10.1017/s002555720000454x|s2cid=124512391 }}==External links==*{{PlanetMath | urlname = QuarticFormula | title = Quartic formula as four single equations }}*[http://members.tripod.com/l_ferrari/quartic_equation.htm Ferrari's achievement]{{Polynomials}}[[Category:Elementary algebra]][[Category:Equations]][[Category:Polynomial functions]]' |
New page wikitext, after the edit ($1) (new_wikitext) | 'theorem is- WHAT IS LOVE BABY DONT HURT ME DONT HURT ME NO MOREEEEEEEEEEE WHAT IS LOVE BABY DONT HURT ME DONT HURT ME NO MOREE WOAHHHHHHH WOAHHH WOAHHH OH OH OH WOAHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH==History==[[Lodovico Ferrari]] is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a [[cubic equation|cubic]] to be found, it could not be published immediately.<ref>{{MacTutor|id=Ferrari|title=Lodovico Ferrari}}</ref> The solution of the quartic was published together with that of the cubic by Ferrari's mentor [[Gerolamo Cardano]] in the book ''[[Ars Magna (Gerolamo Cardano)|Ars Magna]]''.<ref>{{Citation | last = Cardano | first = Gerolamo | author-link = Gerolamo Cardano | year = 1993 | orig-year = 1545 | title = Ars magna or The Rules of Algebra | publisher = Dover | isbn = 0-486-67811-3 | url-access = registration | url = https://archive.org/details/arsmagnaorruleso0000card }}</ref>The Soviet historian I. Y. Depman <small>([[:ru:Депман, Иван Яковлевич|ru]])</small> claimed that even earlier, in 1486, Spanish mathematician Valmes was [[burned at the stake]] for claiming to have solved the quartic equation.<ref>{{citation|last=Depman|title=Rasskazy o matematike|publisher=Gosdetizdat|year=1954|place=Leningrad|language=ru}}</ref> [[Inquisitor General]] [[Tomás de Torquemada]] allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.<ref>{{cite book |author=P. Beckmann |title=A history of π |publisher=Macmillan |year=1971 |page=80 |isbn=9780312381851 |url=https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80}}</ref> However, [[Petr Beckmann]], who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda.<ref>{{cite book |author=P. Beckmann |title=A history of π |publisher=Macmillan |year=1971 |page=191 |isbn=9780312381851 |url=https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80}}</ref> Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed.<ref>{{cite journal|author=P. Zoll | title=Letter to the Editor |journal=American Mathematical Monthly |volume=96 |issue=8 |year=1989 |pages=709–710 |jstor=2324719}}</ref>The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the [[Abel–Ruffini theorem]] in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by [[Évariste Galois]] prior to dying in a duel in 1832 later led to an elegant [[Galois theory|complete theory]] of the roots of polynomials, of which this theorem was one result.<ref>Stewart, Ian, ''Galois Theory, Third Edition'' (Chapman & Hall/CRC Mathematics, 2004)</ref>==Applications==Each [[coordinate]] of the intersection points of two [[conic section]]s is a solution of a quartic equation. The same is true for the intersection of a line and a [[torus]]. It follows that quartic equations often arise in [[computational geometry]] and all related fields such as [[computer graphics]], [[computer-aided design]], [[computer-aided manufacturing]] and [[optics]]. Here are examples of other geometric problems whose solution involves solving a quartic equation.In [[computer-aided manufacturing]], the torus is a shape that is commonly associated with the [[endmill]] cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the {{math|''z''}}-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.<ref>{{Cite web|url=http://people.math.gatech.edu/~etnyre/class/4441Fall16/ShifrinDiffGeo.pdf|title=DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces, p. 36|website=math.gatech.edu}}</ref>A quartic equation arises also in the process of solving the [[crossed ladders problem]], in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Crossed Ladders Problem|url=https://mathworld.wolfram.com/CrossedLaddersProblem.html|access-date=2020-07-27|website=mathworld.wolfram.com|language=en}}</ref>In optics, [[Alhazen's problem]] is "''Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.''" This leads to a quartic equation.<ref name=MacTutor>{{MacTutor|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref><ref>{{citation|title=Scientific Method, Statistical Method and the Speed of Light|first1=R. J.|last1=MacKay|first2=R. W.|last2=Oldford|journal=Statistical Science|volume=15|issue=3|date=August 2000|pages=254–78|doi=10.1214/ss/1009212817|mr=1847825|doi-access=free}}</ref><ref name=Weiss>{{Citation|last = Neumann|first = Peter M.|author-link = Peter M. Neumann|journal = [[American Mathematical Monthly]]|title = Reflections on Reflection in a Spherical Mirror|year = 1998|volume = 105|issue = 6|pages = 523–528|doi = 10.2307/2589403|jstor = 2589403}}</ref>Finding the [[distance of closest approach of ellipses and ellipsoids#Distance of closest approach of two ellipses|distance of closest approach of two ellipses]] involves solving a quartic equation.The [[eigenvalue]]s of a 4×4 [[matrix (mathematics)|matrix]] are the roots of a quartic polynomial which is the [[characteristic polynomial]] of the matrix.The characteristic equation of a fourth-order linear [[difference equation]] or [[differential equation]] is a quartic equation. An example arises in the [[Bending#Timoshenko-Rayleigh theory|Timoshenko-Rayleigh theory]] of beam bending.<ref>{{Cite book|last=Shabana|first=A. A.|url=https://books.google.com/books?id=G2UyBTji18oC&q=Timoshenko-Rayleigh+theory&pg=PA2|title=Theory of Vibration: An Introduction|date=1995-12-08|publisher=Springer Science & Business Media|isbn=978-0-387-94524-8|language=en}}</ref>[[Intersection (Euclidean geometry)|Intersections]] between spheres, cylinders, or other [[quadric]]s can be found using quartic equations.==Inflection points and golden ratio==Letting {{mvar|F}} and {{mvar|G}} be the distinct [[inflection point]]s of the graph of a quartic function, and letting {{mvar|H}} be the intersection of the inflection [[secant line]] {{mvar|FG}} and the quartic, nearer to {{mvar|G}} than to {{mvar|F}}, then {{mvar|G}} divides {{mvar|FH}} into the [[golden section]]:<ref>{{Citation|last = Aude|first = H. T. R.|journal = [[American Mathematical Monthly]]|year = 1949|issue = 3|volume = 56|title = Notes on Quartic Curves|jstor = 2305030|doi = 10.2307/2305030|pages=165–170}}</ref>:<math>\frac{FG}{GH}=\frac{1+\sqrt{5}}{2}= \varphi \; (\text{the golden ratio}).</math>Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.==Solution=====Nature of the roots===Given the general quartic equation:<math>ax^4 + bx^3 + cx^2 + dx + e = 0</math>with real coefficients and {{math|''a'' ≠ 0}} the nature of its roots is mainly determined by the sign of its [[discriminant]] :<math>\begin{align} \Delta = {} &256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 \\ &+ 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e \\&- 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2\end{align} </math> This may be refined by considering the signs of four other polynomials::<math>P = 8ac - 3b^2</math>such that {{math|{{sfrac|''P''|8''a''<sup>2</sup>}}}} is the second degree coefficient of the associated depressed quartic (see [[#Converting_to_a_depressed_quartic|below]]);:<math>R= b^3+8da^2-4abc,</math>such that {{math|{{sfrac|''R''|8''a''<sup>3</sup>}}}} is the first degree coefficient of the associated depressed quartic; :<math>\Delta_0 = c^2 - 3bd + 12ae,</math>which is 0 if the quartic has a triple root; and:<math>D = 64 a^3 e - 16 a^2 c^2 + 16 a b^2 c - 16 a^2 bd - 3 b^4</math>which is 0 if the quartic has two double roots.The possible cases for the nature of the roots are as follows:<ref>{{cite journal|first= E. L.|last=Rees|title=Graphical Discussion of the Roots of a Quartic Equation|journal = The American Mathematical Monthly|volume=29|issue=2|year=1922|pages=51–55|doi=10.2307/2972804|jstor = 2972804}}</ref>* If {{math|∆ < 0}} then the equation has two distinct real roots and two [[complex conjugate]] non-real roots.* If {{math|∆ > 0}} then either the equation's four roots are all real or none is.** If {{mvar|P}} < 0 and {{mvar|D}} < 0 then all four roots are real and distinct.** If {{mvar|P}} > 0 or {{mvar|D}} > 0 then there are two pairs of non-real complex conjugate roots.<ref>{{Cite journal | last1 = Lazard | first1 = D. | doi = 10.1016/S0747-7171(88)80015-4 | title = Quantifier elimination: Optimal solution for two classical examples | journal = Journal of Symbolic Computation | volume = 5 | pages = 261–266 | year = 1988 | issue = 1–2 | doi-access = free }}</ref>* If {{math|∆ {{=}} 0}} then (and only then) the polynomial has a [[multiplicity (mathematics)|multiple]] root. Here are the different cases that can occur:** If {{mvar|P}} < 0 and {{mvar|D}} < 0 and {{math|∆<sub>0</sub> ≠ 0}}, there are a real double root and two real simple roots.** If {{mvar|D}} > 0 or ({{mvar|P}} > 0 and ({{mvar|D}} ≠ 0 or {{mvar|R}} ≠ 0)), there are a real double root and two complex conjugate roots.** If {{math|∆<sub>0</sub> {{=}} 0}} and {{mvar|D}} ≠ 0, there are a triple root and a simple root, all real.** If {{mvar|D}} = 0, then:***If {{mvar|P}} < 0, there are two real double roots.***If {{mvar|P}} > 0 and {{mvar|R}} = 0, there are two complex conjugate double roots.***If {{math|∆<sub>0</sub> {{=}} 0}}, all four roots are equal to {{math|−{{sfrac|''b''|4''a''}}}}There are some cases that do not seem to be covered, but in fact they cannot occur. For example, {{math|∆<sub>0</sub> > 0}}, {{mvar|P}} = 0 and {{mvar|D}} ≤ 0 is not one of the cases. In fact, if {{math|∆<sub>0</sub> > 0}} and {{mvar|P}} = 0 then {{mvar|D}} > 0, since <math>16 a^2\Delta_0 = 3D + P^2; </math> so this combination is not possible.===General formula for roots===[[File:Quartic Formula.svg|thumb|600px|right|Solution of <math>x^4+ax^3+bx^2+cx+d=0</math> written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.]]The four roots {{math|''x''<sub>1</sub>}}, {{math|''x''<sub>2</sub>}}, {{math|''x''<sub>3</sub>}}, and {{math|''x''<sub>4</sub>}} for the general quartic equation:<math>ax^4+bx^3+cx^2+dx+e=0 \,</math>with {{mvar|a}} ≠ 0 are given in the following formula, which is deduced from the one in the section on [[#Ferrari's solution|Ferrari's method]] by back changing the variables (see {{slink||Converting to a depressed quartic}}) and using the formulas for the [[Quadratic function|quadratic]] and [[Cubic function#General formula for roots|cubic equation]]s.:<math>\begin{align}x_{1,2}\ &= -\frac{b}{4a} - S \pm \frac12\sqrt{-4S^2 - 2p + \frac{q}{S}}\\x_{3,4}\ &= -\frac{b}{4a} + S \pm \frac12\sqrt{-4S^2 - 2p - \frac{q}{S}}\end{align}</math>where {{mvar|p}} and {{mvar|q}} are the coefficients of the second and of the first degree respectively in the [[#Converting to a depressed quartic|associated depressed quartic]]:<math>\begin{align}p &= \frac{8ac-3b^2}{8a^2}\\q &= \frac{b^3 - 4abc + 8a^2d}{8a^3} \end{align}</math>:and where:<math>\begin{align}S &= \frac{1}{2}\sqrt{-\frac23\ p+\frac{1}{3a}\left(Q + \frac{\Delta_0}{Q}\right)}\\Q &= \sqrt[3]{\frac{\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2}} \end{align}</math>(if {{math|''S'' {{=}} 0}} or {{math|''Q'' {{=}} 0}}, see {{slink||Special cases of the formula}}, below)with:<math>\begin{align}\Delta_0 &= c^2 - 3bd + 12ae\\\Delta_1 &= 2c^3 - 9bcd + 27b^2 e + 27ad^2 - 72ace\end{align}</math>and:<math>\Delta_1^2-4\Delta_0^3 = - 27 \Delta\ ,</math> where <math>\Delta</math> is the aforementioned [[discriminant]]. For the cube root expression for ''Q'', any of the three cube roots in the complex plane can be used, although if one of them is real that is the natural and simplest one to choose. The mathematical expressions of these last four terms are very similar to those of their [[Cubic function#Algebraic solution|cubic counterparts]].====Special cases of the formula====*If <math>\Delta > 0,</math> the value of <math>Q</math> is a non-real complex number. In this case, either all roots are non-real or they are all real. In the latter case, the value of <math>S</math> is also real, despite being expressed in terms of <math>Q;</math> this is [[casus irreducibilis]] of the cubic function extended to the present context of the quartic. One may prefer to express it in a purely real way, by using [[trigonometric functions]], as follows:::<math>S = \frac{1}{2} \sqrt{-\frac23\ p+\frac{2}{3a}\sqrt{\Delta_0}\cos\frac{\varphi}{3}}</math>:where::<math>\varphi = \arccos\left(\frac{\Delta_1}{2\sqrt{\Delta_0^3}}\right).</math>*If <math>\Delta \neq 0</math> and <math>\Delta_0 = 0,</math> the sign of <math>\sqrt{\Delta_1^2 - 4 \Delta_0^3}=\sqrt{\Delta_1^2} </math> has to be chosen to have <math>Q \neq 0,</math> that is one should define <math>\sqrt{\Delta_1^2}</math> as <math>\Delta_1,</math> maintaining the sign of <math>\Delta_1.</math>*If <math>S = 0,</math> then one must change the choice of the cube root in <math>Q</math> in order to have <math>S \neq 0.</math> This is always possible except if the quartic may be factored into <math>\left(x+\tfrac{b}{4a}\right)^4.</math> The result is then correct, but misleading because it hides the fact that no cube root is needed in this case. In fact this case{{Clarify|reason=Both kinds or specific?|date=March 2024}} may occur only if the [[numerator]] of <math>q</math> is zero, in which case the associated [[#Converting to a depressed quartic|depressed quartic]] is biquadratic; it may thus be solved by the method described [[#Biquadratic equation|below]].*If <math>\Delta = 0</math> and <math>\Delta_0 = 0,</math> and thus also <math>\Delta_1 = 0,</math> at least three roots are equal to each other, and the roots are [[rational function]]s of the coefficients. The triple root <math>x_0</math> is a common root of the quartic and its second derivative <math>2(6ax^2+3bx+c);</math> it is thus also the unique root of the remainder of the [[Euclidean division]] of the quartic by its second derivative, which is a linear polynomial. The simple root <math>x_1</math> can be deduced from <math>x_1+3x_0=-b/a.</math>*If <math>\Delta=0</math> and <math> \Delta_0 \neq 0,</math> the above expression for the roots is correct but misleading, hiding the fact that the polynomial is [[irreducible polynomial|reducible]] and no cube root is needed to represent the roots.===Simpler cases=======Reducible quartics====Consider the general quartic:<math>Q(x) = a_4x^4+a_3x^3+a_2x^2+a_1x+a_0.</math>It is [[irreducible polynomial|reducible]] if {{math|''Q''(''x'') {{=}} ''R''(''x'')×''S''(''x'')}}, where {{math|''R''(''x'')}} and {{math|''S''(''x'')}} are non-constant polynomials with [[rational number|rational]] coefficients (or more generally with coefficients in the same [[field (mathematics)|field]] as the coefficients of {{math|''Q''(''x'')}}). Such a factorization will take one of two forms::<math>Q(x) = (x-x_1)(b_3x^3+b_2x^2+b_1x+b_0)</math>or:<math>Q(x) = (c_2x^2+c_1x+c_0)(d_2x^2+d_1x+d_0).</math>In either case, the roots of {{math|''Q''(''x'')}} are the roots of the factors, which may be computed using the formulas for the roots of a [[quadratic function]] or [[cubic function]].Detecting the existence of such factorizations can be done [[Resolvent cubic#Factoring quartic polynomials|using the resolvent cubic of {{math|''Q''(''x'')}}]]. It turns out that:* if we are working over {{math|'''R'''}} (that is, if coefficients are restricted to be real numbers) (or, more generally, over some [[real closed field]]) then there is always such a factorization;* if we are working over {{math|'''Q'''}} (that is, if coefficients are restricted to be rational numbers) then there is an algorithm to determine whether or not {{math|''Q''(''x'')}} is reducible and, if it is, how to express it as a product of polynomials of smaller degree.In fact, several methods of solving quartic equations ([[Quartic function#Ferrari's solution|Ferrari's method]], [[Quartic function#Descartes' solution|Descartes' method]], and, to a lesser extent, [[Quartic function#Euler's solution|Euler's method]]) are based upon finding such factorizations.====Biquadratic equation====If {{math|''a''<sub>3</sub> {{=}} ''a''<sub>1</sub> {{=}} 0}} then the function :<math>Q(x) = a_4x^4+a_2x^2+a_0</math> is called a '''biquadratic function'''; equating it to zero defines a '''biquadratic equation''', which is easy to solve as followsLet the auxiliary variable {{math|''z'' {{=}} ''x''<sup>2</sup>}}.Then {{math|''Q''(''x'')}} becomes a [[Quadratic function|quadratic]] {{math|''q''}} in {{math|''z''}}: {{math|''q''(''z'') {{=}} ''a''<sub>4</sub>''z''<sup>2</sup> + ''a''<sub>2</sub>''z'' + ''a''<sub>0</sub>}}. Let {{math|''z''<sub>+</sub>}} and {{math|''z''<sub>−</sub>}} be the roots of {{math|''q''(''z'')}}. Then the roots of the quartic {{math|''Q''(''x'')}} are:<math>\begin{align}x_1&=+\sqrt{z_+},\\x_2&=-\sqrt{z_+},\\x_3&=+\sqrt{z_-},\\x_4&=-\sqrt{z_-}.\end{align}</math>==== Quasi-palindromic equation ====The polynomial: <math>P(x)=a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2</math>is almost [[Reciprocal polynomial#Palindromic polynomial|palindromic]], as {{math|''P''(''mx'') {{=}} {{sfrac|''x''<sup>4</sup>|''m''<sup>2</sup>}}''P''({{sfrac|''m''|''x''}})}} (it is palindromic if {{math|''m'' {{=}} 1}}). The change of variables {{math|''z'' {{=}} ''x'' + {{sfrac|''m''|''x''}}}} in {{math|{{sfrac|''P''(''x'')|''x''<sup>2</sup>}} {{=}} 0}} produces the [[quadratic equation]] {{math|''a''<sub>0</sub>''z''<sup>2</sup> + ''a''<sub>1</sub>''z'' + ''a''<sub>2</sub> − 2''ma''<sub>0</sub> {{=}} 0}}. Since {{math|''x''<sup>2</sup> − ''xz'' + ''m'' {{=}} 0}}, the quartic equation {{math|''P''(''x'') {{=}} 0}} may be solved by applying the [[quadratic formula]] twice.===Solution methods=======Converting to a depressed quartic====For solving purposes, it is generally better to convert the quartic into a '''depressed quartic''' by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.Let:<math> a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0 </math>be the general quartic equation we want to solve.Dividing by {{math|''a''<sub>4</sub>}}, provides the equivalent equation {{math|''x''<sup>4</sup> + ''bx''<sup>3</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e'' {{=}} 0}}, with {{math|''b'' {{=}} {{sfrac|''a''<sub>3</sub>|''a''<sub>4</sub>}}}}, {{math|''c'' {{=}} {{sfrac|''a''<sub>2</sub>|''a''<sub>4</sub>}}}}, {{math|''d'' {{=}} {{sfrac|''a''<sub>1</sub>|''a''<sub>4</sub>}}}}, and {{math|''e'' {{=}} {{sfrac|''a''<sub>0</sub>|''a''<sub>4</sub>}}}}.Substituting {{math|''y'' − {{sfrac|''b''|4}}}} for {{mvar|x}} gives, after regrouping the terms, the equation {{math|''y''<sup>4</sup> + ''py''<sup>2</sup> + ''qy'' + ''r'' {{=}} 0}},where:<math>\begin{align}p&=\frac{8c-3b^2}{8} =\frac{8a_2a_4-3{a_3}^2}{8{a_4}^2}\\q&=\frac{b^3-4bc+8d}{8} =\frac{{a_3}^3-4a_2a_3a_4+8a_1{a_4}^2}{8{a_4}^3}\\r&=\frac{-3b^4+256e-64bd+16b^2c}{256}=\frac{-3{a_3}^4+256a_0{a_4}^3-64a_1a_3{a_4}^2+16a_2{a_3}^2a_4}{256{a_4}^4}.\end{align}</math>If {{math|''y''<sub>0</sub>}} is a root of this depressed quartic, then {{math|''y''<sub>0</sub> − {{sfrac|''b''|4}}}} (that is {{math|''y''<sub>0</sub> − {{sfrac|''a''<sub>3</sub>|4''a''<sub>4</sub>}})}} is a root of the original quartic and every root of the original quartic can be obtained by this process.====Ferrari's solution====As explained in the preceding section, we may start with the ''depressed quartic equation'':<math> y^4 + p y^2 + q y + r = 0. </math>This depressed quartic can be solved by means of a method discovered by [[Lodovico Ferrari]]. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as:<math> \left(y^2 + \frac p2\right)^2 = -q y - r + \frac{p^2}4. </math>Then, we introduce a variable {{mvar|m}} into the factor on the left-hand side by adding {{math|2''y''<sup>2</sup>''m'' + ''pm'' + ''m''<sup>2</sup>}} to both sides. After regrouping the coefficients of the power of {{mvar|y}} on the right-hand side, this gives the equation {{NumBlk|:|<math> \left(y^2 + \frac p2 + m\right)^2 = 2 m y^2 - q y + m^2 + m p + \frac{p^2}4 - r, </math>|{{EquationRef|1}}}}which is equivalent to the original equation, whichever value is given to {{mvar|m}}.As the value of {{mvar|m}} may be arbitrarily chosen, we will choose it in order to [[complete the square]] on the right-hand side. This implies that the [[discriminant]] in {{mvar|y}} of this [[quadratic equation]] is zero, that is {{mvar|m}} is a root of the equation:<math> (-q)^2 - 4 (2m)\left(m^2 + p m + \frac{p^2}4 - r\right) = 0,\,</math>which may be rewritten as{{NumBlk|:|<math>8m^3+ 8pm^2 + (2p^2 -8r)m- q^2 =0.</math>|{{EquationRef|1a}}}}This is the [[resolvent cubic]] of the quartic equation. The value of {{mvar|m}} may thus be obtained from [[Cubic equation#Cardano's method|Cardano's formula]]. When {{mvar|m}} is a root of this equation, the right-hand side of equation (''{{EquationNote|1}}'') is the square:<math>\left(\sqrt{2m}y-\frac q{2\sqrt{2m}}\right)^2.</math>However, this induces a division by zero if {{math|''m'' {{=}} 0}}. This implies {{math|''q'' {{=}} 0}}, and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that {{math|''m'' ≠ 0}}. This is always possible except for the depressed equation {{math|''y''<sup>4</sup> {{=}} 0}}.Now, if {{mvar|m}} is a root of the cubic equation such that {{math|''m'' ≠ 0}}, equation (''{{EquationNote|1}}'') becomes:<math> \left(y^2 + \frac p2 + m\right)^2 = \left(y\sqrt{2 m}-\frac{q}{2\sqrt{2 m}}\right)^2. </math>This equation is of the form {{math|''M''<sup>2</sup> {{=}} ''N''<sup>2</sup>}}, which can be rearranged as {{math|''M''<sup>2</sup> − ''N''<sup>2</sup> {{=}} 0}} or {{math|(''M'' + ''N'')(''M'' − ''N'') {{=}} 0}}. Therefore, equation (''{{EquationNote|1}}'') may be rewritten as:<math> \left(y^2 + \frac p2 + m + \sqrt{2 m}y-\frac q{2\sqrt{2 m}}\right) \left(y^2 + \frac p2 + m - \sqrt{2 m}y+\frac q{2\sqrt{2 m}}\right)=0.</math>This equation is easily solved by applying to each factor the [[quadratic formula]]. Solving them we may write the four roots as:<math>y={\pm_1\sqrt{2 m} \pm_2 \sqrt{-\left(2p + 2m \pm_1 {\sqrt 2q \over \sqrt{m}} \right)} \over 2},</math>where {{math|±<sub>1</sub>}} and {{math|±<sub>2</sub>}} denote either {{math|+}} or {{math|−}}. As the two occurrences of {{math|±<sub>1</sub>}} must denote the same sign, this leaves four possibilities, one for each root.Therefore, the solutions of the original quartic equation are:<math>x=-{a_3 \over 4a_4} + {\pm_1\sqrt{2 m} \pm_2 \sqrt{-\left(2p + 2m \pm_1 {\sqrt2q \over \sqrt{m}} \right)} \over 2}.</math> A comparison with the [[#General_formula_for_roots|general formula]] above shows that {{math|{{sqrt|2''m''}} {{=}} 2''S''}}.====Descartes' solution====Descartes<ref>{{Citation|last = Descartes|first = René|author-link = René Descartes|title = [[La Géométrie|The Geometry of Rene Descartes with a facsimile of the first edition]]|isbn = 0-486-60068-8|publisher = [[Dover Publications|Dover]]|year = 1954|jfm = 51.0020.07|chapter = Book III: On the construction of solid and supersolid problems|orig-year = 1637}}</ref> introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let:<math> \begin{align} x^4 + bx^3 + cx^2 + dx + e & = (x^2 + sx + t)(x^2 + ux + v) \\ & = x^4 + (s + u)x^3 + (t + v + su)x^2 + (sv + tu)x + tv \end{align} </math>By [[equating coefficients]], this results in the following system of equations::<math> \left\{\begin{array}{l} b = s + u \\ c = t + v + su \\ d = sv + tu \\ e = tv \end{array}\right. </math>This can be simplified by starting again with the [[#Converting to a depressed quartic|depressed quartic]] {{math|''y''<sup>4</sup> + ''py''<sup>2</sup> + ''qy'' + ''r''}}, which can be obtained by substituting {{math|''y'' − ''b''/4}} for {{math|''x''}}. Since the coefficient of {{math|''y''<sup>3</sup>}} is {{math|0}}, we get {{math|''s'' {{=}} −''u''}}, and::<math> \left\{\begin{array}{l} p + u^2 = t + v \\ q = u (t - v) \\ r = tv \end{array}\right. </math>One can now eliminate both {{mvar|t}} and {{mvar|v}} by doing the following::<math> \begin{align} u^2(p + u^2)^2 - q^2 & = u^2(t + v)^2 - u^2(t - v)^2 \\ & = u^2 [(t + v + (t - v))(t + v - (t - v))]\\ & = u^2(2t)(2v) \\ & = 4u^2tv \\ & = 4u^2r \end{align} </math>If we set {{math|''U'' {{=}} ''u''<sup>2</sup>}}, then solving this equation becomes finding the roots of the [[resolvent cubic]]{{NumBlk|:|<math> U^3 + 2pU^2 + (p^2-4r)U - q^2,</math>|{{EquationRef|2}}}}which is [[Cubic function#General solution to the cubic equation with real coefficients|done elsewhere]]. This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m.If {{math|''u''}} is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic {{math|''x''<sup>4</sup>}}, which is trivially factored),:<math> \left\{\begin{array}{l} s = -u \\ 2t = p + u^2 + q/u \\ 2v = p + u^2 - q/u \end{array}\right. </math>The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of {{mvar|u}} for the square root of {{mvar|U}} merely exchanges the two quadratics with one another.The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic (''{{EquationNote|2}}'') has a non-zero root which is the square of a rational, or {{math|''p''<sup>2</sup> − 4''r''}} is the square of rational and {{math|''q'' {{=}} 0}}; this can readily be checked using the [[rational root test]].<ref name=Brookfield>{{cite journal |author=Brookfield, G. |title=Factoring quartic polynomials: A lost art |journal=[[Mathematics Magazine]] |volume=80 |issue=1 |year=2007 |pages=67–70|doi=10.1080/0025570X.2007.11953453 |s2cid=53375377 |url = https://www.maa.org/sites/default/files/Brookfield2007-103574.pdf}}</ref>====Euler's solution====A variant of the previous method is due to [[Leonhard Euler|Euler]].<ref>{{Citation|last = van der Waerden|first=Bartel Leendert|author-link = Bartel Leendert van der Waerden|title = [[Moderne Algebra|Algebra]]|volume = 1|publisher=[[Springer Science+Business Media|Springer-Verlag]]|edition = 7th|isbn = 0-387-97424-5|year = 1991|section = The Galois theory: Equations of the second, third, and fourth degrees|zbl = 0724.12001}}</ref><ref>{{Citation|last = Euler|first = Leonhard|author-link = Leonhard Euler|title = [[Elements of Algebra]]|chapter= Of a new method of resolving equations of the fourth degree|publisher=[[Springer Science+Business Media|Springer-Verlag]]|orig-year = 1765|year = 1984|zbl = 0557.01014|isbn = 978-1-4613-8511-0}}</ref> Unlike the previous methods, both of which use ''some'' root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r''}}. Observe that, if* {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r'' {{=}} (''x''<sup>2</sup> + ''sx'' + ''t'')(''x''<sup>2</sup> − ''sx'' + ''v'')}},* {{math|''r''<sub>1</sub>}} and {{math|''r''<sub>2</sub>}} are the roots of {{math|''x''<sup>2</sup> + ''sx'' + ''t''}},* {{math|''r''<sub>3</sub>}} and {{math|''r''<sub>4</sub>}} are the roots of {{math|''x''<sup>2</sup> − ''sx'' + ''v''}},then* the roots of {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r''}} are {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}},* {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> {{=}} −''s''}},* {{math|''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} ''s''}}.Therefore, {{math|(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>) {{=}} −''s''<sup>2</sup>}}. In other words, {{math|−(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>)}} is one of the roots of the resolvent cubic (''{{EquationNote|2}}'') and this suggests that the roots of that cubic are equal to {{math|−(''r''<sub>1</sub> + ''r''<sub>2</sub>)(''r''<sub>3</sub> + ''r''<sub>4</sub>)}}, {{math|−(''r''<sub>1</sub> + ''r''<sub>3</sub>)(''r''<sub>2</sub> + ''r''<sub>4</sub>)}}, and {{math|−(''r''<sub>1</sub> + ''r''<sub>4</sub>)(''r''<sub>2</sub> + ''r''<sub>3</sub>)}}. This is indeed true and it follows from [[Vieta's formulas]]. It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> + ''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} 0}}. (Of course, this also follows from the fact that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub> + ''r''<sub>3</sub> + ''r''<sub>4</sub> {{=}} −''s'' + ''s''}}.) Therefore, if {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are the roots of the resolvent cubic, then the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} are such that:<math>\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\(r_1+r_2)(r_3+r_4)=-\alpha\\(r_1+r_3)(r_2+r_4)=-\beta\\(r_1+r_4)(r_2+r_3)=-\gamma\text{.}\end{array}\right.</math>It is a consequence of the first two equations that {{math|''r''<sub>1</sub> + ''r''<sub>2</sub>}} is a square root of {{math|''α''}} and that {{math|''r''<sub>3</sub> + ''r''<sub>4</sub>}} is the other square root of {{math|''α''}}. For the same reason,* {{math|''r''<sub>1</sub> + ''r''<sub>3</sub>}} is a square root of {{math|''β''}},* {{math|''r''<sub>2</sub> + ''r''<sub>4</sub>}} is the other square root of {{math|''β''}},* {{math|''r''<sub>1</sub> + ''r''<sub>4</sub>}} is a square root of {{math|''γ''}},* {{math|''r''<sub>2</sub> + ''r''<sub>3</sub>}} is the other square root of {{math|''γ''}}.Therefore, the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} are such that:<math>\left\{\begin{array}{l}r_1+r_2+r_3+r_4=0\\r_1+r_2=\sqrt{\alpha}\\r_1+r_3=\sqrt{\beta}\\r_1+r_4=\sqrt{\gamma}\text{;}\end{array}\right.</math>the sign of the square roots will be dealt with below. The only solution of this system is::<math>\left\{\begin{array}{l}r_1=\frac{\sqrt{\alpha}+\sqrt{\beta}+\sqrt{\gamma}}2\\[2mm]r_2=\frac{\sqrt{\alpha}-\sqrt{\beta}-\sqrt{\gamma}}2\\[2mm]r_3=\frac{-\sqrt{\alpha}+\sqrt{\beta}-\sqrt{\gamma}}2\\[2mm]r_4=\frac{-\sqrt{\alpha}-\sqrt{\beta}+\sqrt{\gamma}}2\text{.}\end{array}\right.</math>Since, in general, there are two choices for each square root, it might look as if this provides {{math|8 ({{=}} 2<sup>3</sup>)}} choices for the set {{math|{''r''<sub>1</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, ''r''<sub>4</sub>}}}, but, in fact, it provides no more than {{math|2}} such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set {{math|{''r''<sub>1</sub>, ''r''<sub>2</sub>, ''r''<sub>3</sub>, ''r''<sub>4</sub>}}} becomes the set {{math|{−''r''<sub>1</sub>, −''r''<sub>2</sub>, −''r''<sub>3</sub>, −''r''<sub>4</sub>}}}.In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} and uses them to compute the numbers {{math|''r''<sub>1</sub>}}, {{math|''r''<sub>2</sub>}}, {{math|''r''<sub>3</sub>}}, and {{math|''r''<sub>4</sub>}} from the previous equalities. Then, one computes the number {{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}}}}. Since {{math|''α''}}, {{math|''β''}}, and {{math|''γ''}} are the roots of (''{{EquationNote|2}}''), it is a consequence of Vieta's formulas that their product is equal to {{math|''q''<sup>2</sup>}} and therefore that {{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}} {{=}} ±''q''}}. But a straightforward computation shows that:{{math|{{sqrt|''α''}}{{sqrt|''β''}}{{sqrt|''γ''}} {{=}} ''r''<sub>1</sub>''r''<sub>2</sub>''r''<sub>3</sub> + ''r''<sub>1</sub>''r''<sub>2</sub>''r''<sub>4</sub> + ''r''<sub>1</sub>''r''<sub>3</sub>''r''<sub>4</sub> + ''r''<sub>2</sub>''r''<sub>3</sub>''r''<sub>4</sub>.}}If this number is {{math|−''q''}}, then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be {{math|−''r''<sub>1</sub>}}, {{math|−''r''<sub>2</sub>}}, {{math|−''r''<sub>3</sub>}}, and {{math|−''r''<sub>4</sub>}}, which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one).This argument suggests another way of choosing the square roots:* pick ''any'' square root {{math|{{sqrt|''α''}}}} of {{math|''α''}} and ''any'' square root {{math|{{sqrt|''β''}}}} of {{math|''β''}};* ''define'' {{math|{{sqrt|''γ''}}}} as <math>-\frac q{\sqrt{\alpha}\sqrt{\beta}}</math>.Of course, this will make no sense if {{math|''α''}} or {{math|''β''}} is equal to {{math|0}}, but {{math|0}} is a root of (''{{EquationNote|2}}'') only when {{math|''q'' {{=}} 0}}, that is, only when we are dealing with a [[Quartic function#Biquadratic equation|biquadratic equation]], in which case there is a much simpler approach.====Solving by Lagrange resolvent====The [[symmetric group]] {{math|''S''<sub>4</sub>}} on four elements has the [[Klein four-group]] as a [[normal subgroup]]. This suggests using a '''{{visible anchor|resolvent cubic}}''' whose roots may be variously described as a discrete Fourier transform or a [[Hadamard matrix]] transform of the roots; see [[Lagrange resolvents]] for the general method. Denote by {{math|''x<sub>i</sub>''}}, for {{math|''i''}} from {{math|0}} to {{math|3}}, the four roots of {{math|''x''<sup>4</sup> + ''bx''<sup>3</sup> + ''cx''<sup>2</sup> + ''dx'' + ''e''}}. If we set: <math> \begin{align}s_0 &= \tfrac12(x_0 + x_1 + x_2 + x_3), \\[4pt]s_1 &= \tfrac12(x_0 - x_1 + x_2 - x_3), \\[4pt]s_2 &= \tfrac12(x_0 + x_1 - x_2 - x_3), \\[4pt]s_3 &= \tfrac12(x_0 - x_1 - x_2 + x_3),\end{align}</math>then since the transformation is an [[Involution (mathematics)|involution]] we may express the roots in terms of the four {{math|''s<sub>i</sub>''}} in exactly the same way. Since we know the value {{math|''s''<sub>0</sub> {{=}} −{{sfrac|''b''|2}}}}, we only need the values for {{math|''s''<sub>1</sub>}}, {{math|''s''<sub>2</sub>}} and {{math|''s''<sub>3</sub>}}. These are the roots of the polynomial:<math>(s^2 - {s_1}^2)(s^2-{s_2}^2)(s^2-{s_3}^2).</math>Substituting the {{math|''s<sub>i</sub>''}} by their values in term of the {{math|''x<sub>i</sub>''}}, this polynomial may be expanded in a polynomial in {{math|''s''}} whose coefficients are [[symmetric polynomial]]s in the {{math|''x<sub>i</sub>''}}. By the [[fundamental theorem of symmetric polynomials]], these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is {{math|''b'' {{=}} 0}}, this results in the polynomial{{NumBlk|:|<math> s^6+2cs^4+(c^2-4e)s^2-d^2 </math>|{{EquationRef|3}}}}This polynomial is of degree six, but only of degree three in {{math|''s''<sup>2</sup>}}, and so the corresponding equation is solvable by the method described in the article about [[cubic function]]. By substituting the roots in the expression of the {{math|''x<sub>i</sub>''}} in terms of the {{math|''s<sub>i</sub>''}}, we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the {{math|''x<sub>i</sub>''}}.These expressions are unnecessarily complicated, involving the [[root of unity|cubic roots of unity]], which can be avoided as follows. If {{math|''s''}} is any non-zero root of (''{{EquationNote|3}}''), and if we set:<math>\begin{align}F_1(x) & = x^2 + sx + \frac{c}{2} + \frac{s^2}{2} - \frac{d}{2s} \\F_2(x) & = x^2 - sx + \frac{c}{2} + \frac{s^2}{2} + \frac{d}{2s}\end{align}</math>then:<math>F_1(x)\times F_2(x) = x^4 + cx^2 + dx + e.</math>We therefore can solve the quartic by solving for {{math|''s''}} and then solving for the roots of the two factors using the [[quadratic formula]].This gives exactly the same formula for the roots as the one provided by [[Quartic function#Descartes' solution|Descartes' method]].====Solving with algebraic geometry====There is an alternative solution using algebraic geometry<ref>{{Citation|last = Faucette|first = William M.|journal = [[American Mathematical Monthly]]|pages = 51–57|title = A Geometric Interpretation of the Solution of the General Quartic Polynomial|volume = 103|year = 1996|issue = 1|doi = 10.2307/2975214|jstor = 2975214|mr = 1369151}}</ref> In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three [[degenerate conic|reducible quadratic curves]] (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic.The four roots of the depressed quartic {{math|''x''<sup>4</sup> + ''px''<sup>2</sup> + ''qx'' + ''r'' {{=}} 0}} may also be expressed as the {{mvar|x}} coordinates of the intersections of the two quadratic equations {{math|''y''<sup>2</sup> + ''py'' + ''qx'' + ''r'' {{=}} 0}} and {{math|''y'' − ''x''<sup>2</sup> {{=}} 0}} i.e., using the substitution {{math|''y'' {{=}} ''x''<sup>2</sup>}} that two quadratics intersect in four points is an instance of [[Bézout's theorem]]. Explicitly, the four points are {{math|''P<sub>i</sub>'' ≔ (''x<sub>i</sub>'', ''x<sub>i</sub>''<sup>2</sup>)}} for the four roots {{math|''x<sub>i</sub>''}} of the quartic.These four points are not collinear because they lie on the irreducible quadratic {{math|''y'' {{=}} ''x''<sup>2</sup>}} and thus there is a 1-parameter family of quadratics (a [[pencil of curves]]) passing through these points. Writing the projectivization of the two quadratics as [[quadratic form]]s in three variables::<math>\begin{align}F_1(X,Y,Z) &:= Y^2 + pYZ + qXZ + rZ^2,\\F_2(X,Y,Z) &:= YZ - X^2\end{align}</math>the pencil is given by the forms {{math|''λF''<sub>1</sub> + ''μF''<sub>2</sub>}} for any point {{math|[''λ'', ''μ'']}} in the projective line — in other words, where {{math|''λ''}} and {{math|''μ''}} are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros.This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done <math>\textstyle{\binom{4}{2}}</math> = {{math|6}} different ways. Denote these {{math|''Q''<sub>1</sub> {{=}} ''L''<sub>12</sub> + ''L''<sub>34</sub>}}, {{math|''Q''<sub>2</sub> {{=}} ''L''<sub>13</sub> + ''L''<sub>24</sub>}}, and {{math|''Q''<sub>3</sub> {{=}} ''L''<sub>14</sub> + ''L''<sub>23</sub>}}. Given any two of these, their intersection has exactly the four points.The reducible quadratics, in turn, may be determined by expressing the quadratic form {{math|''λF''<sub>1</sub> + ''μF''<sub>2</sub>}} as a {{math|3×3}} matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in {{math|''λ''}} and {{math|''μ''}} and corresponds to the resolvent cubic.==See also==*{{annotated link|Linear function}}*{{annotated link|Quadratic function}}*{{annotated link|Cubic function}}*{{annotated link|Quintic function}}==Notes==:{{note|Alpha|α}} For the purposes of this article, ''e'' is used as a [[Variable (mathematics)|variable]] as opposed to its conventional use as [[e (mathematical constant)|Euler's number]](except when otherwise specified).==References=={{reflist}}==Further reading==* {{cite journal |author=Carpenter, W. |title=On the solution of the real quartic |journal=[[Mathematics Magazine]] |volume=39 |year=1966 |issue=1 |pages=28–30 |doi=10.2307/2688990|jstor=2688990 }}* {{cite journal |author1=Yacoub, M.D.|author2=Fraidenraich, G. |title=A solution to the quartic equation |journal=[[Mathematical Gazette]] |volume=96|date=July 2012|pages=271–275 |doi=10.1017/s002555720000454x|s2cid=124512391 }}==External links==*{{PlanetMath | urlname = QuarticFormula | title = Quartic formula as four single equations }}*[http://members.tripod.com/l_ferrari/quartic_equation.htm Ferrari's achievement]{{Polynomials}}[[Category:Elementary algebra]][[Category:Equations]][[Category:Polynomial functions]]' |
Parsed HTML source of the new revision ($1) (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>theorem is- WHAT IS LOVE BABY DONT HURT ME DONT HURT ME NO MOREEEEEEEEEEE WHAT IS LOVE BABY DONT HURT ME DONT HURT ME NO MOREE WOAHHHHHHH WOAHHH WOAHHH OH OH OH WOAHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH</p><div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div><ul><li class="toclevel-1 tocsection-1"><a href="#History"><span class="tocnumber">1</span> <span class="toctext">History</span></a></li><li class="toclevel-1 tocsection-2"><a href="#Applications"><span class="tocnumber">2</span> <span class="toctext">Applications</span></a></li><li class="toclevel-1 tocsection-3"><a href="#Inflection_points_and_golden_ratio"><span class="tocnumber">3</span> <span class="toctext">Inflection points and golden ratio</span></a></li><li class="toclevel-1 tocsection-4"><a href="#Solution"><span class="tocnumber">4</span> <span class="toctext">Solution</span></a><ul><li class="toclevel-2 tocsection-5"><a href="#Nature_of_the_roots"><span class="tocnumber">4.1</span> <span class="toctext">Nature of the roots</span></a></li><li class="toclevel-2 tocsection-6"><a href="#General_formula_for_roots"><span class="tocnumber">4.2</span> <span class="toctext">General formula for roots</span></a><ul><li class="toclevel-3 tocsection-7"><a href="#Special_cases_of_the_formula"><span class="tocnumber">4.2.1</span> <span class="toctext">Special cases of the formula</span></a></li></ul></li><li class="toclevel-2 tocsection-8"><a href="#Simpler_cases"><span class="tocnumber">4.3</span> <span class="toctext">Simpler cases</span></a><ul><li class="toclevel-3 tocsection-9"><a href="#Reducible_quartics"><span class="tocnumber">4.3.1</span> <span class="toctext">Reducible quartics</span></a></li><li class="toclevel-3 tocsection-10"><a href="#Biquadratic_equation"><span class="tocnumber">4.3.2</span> <span class="toctext">Biquadratic equation</span></a></li><li class="toclevel-3 tocsection-11"><a href="#Quasi-palindromic_equation"><span class="tocnumber">4.3.3</span> <span class="toctext">Quasi-palindromic equation</span></a></li></ul></li><li class="toclevel-2 tocsection-12"><a href="#Solution_methods"><span class="tocnumber">4.4</span> <span class="toctext">Solution methods</span></a><ul><li class="toclevel-3 tocsection-13"><a href="#Converting_to_a_depressed_quartic"><span class="tocnumber">4.4.1</span> <span class="toctext">Converting to a depressed quartic</span></a></li><li class="toclevel-3 tocsection-14"><a href="#Ferrari's_solution"><span class="tocnumber">4.4.2</span> <span class="toctext">Ferrari's solution</span></a></li><li class="toclevel-3 tocsection-15"><a href="#Descartes'_solution"><span class="tocnumber">4.4.3</span> <span class="toctext">Descartes' solution</span></a></li><li class="toclevel-3 tocsection-16"><a href="#Euler's_solution"><span class="tocnumber">4.4.4</span> <span class="toctext">Euler's solution</span></a></li><li class="toclevel-3 tocsection-17"><a href="#Solving_by_Lagrange_resolvent"><span class="tocnumber">4.4.5</span> <span class="toctext">Solving by Lagrange resolvent</span></a></li><li class="toclevel-3 tocsection-18"><a href="#Solving_with_algebraic_geometry"><span class="tocnumber">4.4.6</span> <span class="toctext">Solving with algebraic geometry</span></a></li></ul></li></ul></li><li class="toclevel-1 tocsection-19"><a href="#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li><li class="toclevel-1 tocsection-20"><a href="#Notes"><span class="tocnumber">6</span> <span class="toctext">Notes</span></a></li><li class="toclevel-1 tocsection-21"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li><li class="toclevel-1 tocsection-22"><a href="#Further_reading"><span class="tocnumber">8</span> <span class="toctext">Further reading</span></a></li><li class="toclevel-1 tocsection-23"><a href="#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a></li></ul></div><h2><span class="mw-headline" id="History">History</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=1" title="Edit section's source code: History"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><p><a href="https://www.search.com.vn/wiki/en/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a> is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a <a href="https://www.search.com.vn/wiki/en/Cubic_equation" title="Cubic equation">cubic</a> to be found, it could not be published immediately.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> The solution of the quartic was published together with that of the cubic by Ferrari's mentor <a href="https://www.search.com.vn/wiki/en/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> in the book <i><a href="https://www.search.com.vn/wiki/en/Ars_Magna_(Gerolamo_Cardano)" class="mw-redirect" title="Ars Magna (Gerolamo Cardano)">Ars Magna</a></i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup></p><p>The Soviet historian I. Y. Depman <small>(<a href="https://www.search.com.vn/wiki/ru/%D0%94%D0%B5%D0%BF%D0%BC%D0%B0%D0%BD,_%D0%98%D0%B2%D0%B0%D0%BD_%D0%AF%D0%BA%D0%BE%D0%B2%D0%BB%D0%B5%D0%B2%D0%B8%D1%87" class="extiw" title="ru:Депман, Иван Яковлевич">ru</a>)</small> claimed that even earlier, in 1486, Spanish mathematician Valmes was <a href="https://www.search.com.vn/wiki/en/Burned_at_the_stake" class="mw-redirect" title="Burned at the stake">burned at the stake</a> for claiming to have solved the quartic equation.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup> <a href="https://www.search.com.vn/wiki/en/Inquisitor_General" class="mw-redirect" title="Inquisitor General">Inquisitor General</a> <a href="https://www.search.com.vn/wiki/en/Tom%C3%A1s_de_Torquemada" title="Tomás de Torquemada">Tomás de Torquemada</a> allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> However, <a href="https://www.search.com.vn/wiki/en/Petr_Beckmann" title="Petr Beckmann">Petr Beckmann</a>, who popularized this story of Depman in the West, said that it was unreliable and hinted that it may have been invented as Soviet antireligious propaganda.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup> Beckmann's version of this story has been widely copied in several books and internet sites, usually without his reservations and sometimes with fanciful embellishments. Several attempts to find corroborating evidence for this story, or even for the existence of Valmes, have failed.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6">[6]</a></sup></p><p>The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the <a href="https://www.search.com.vn/wiki/en/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by <a href="https://www.search.com.vn/wiki/en/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> prior to dying in a duel in 1832 later led to an elegant <a href="https://www.search.com.vn/wiki/en/Galois_theory" title="Galois theory">complete theory</a> of the roots of polynomials, of which this theorem was one result.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">[7]</a></sup></p><h2><span class="mw-headline" id="Applications">Applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=2" title="Edit section's source code: Applications"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><p>Each <a href="https://www.search.com.vn/wiki/en/Coordinate" class="mw-redirect" title="Coordinate">coordinate</a> of the intersection points of two <a href="https://www.search.com.vn/wiki/en/Conic_section" title="Conic section">conic sections</a> is a solution of a quartic equation. The same is true for the intersection of a line and a <a href="https://www.search.com.vn/wiki/en/Torus" title="Torus">torus</a>. It follows that quartic equations often arise in <a href="https://www.search.com.vn/wiki/en/Computational_geometry" title="Computational geometry">computational geometry</a> and all related fields such as <a href="https://www.search.com.vn/wiki/en/Computer_graphics" title="Computer graphics">computer graphics</a>, <a href="https://www.search.com.vn/wiki/en/Computer-aided_design" title="Computer-aided design">computer-aided design</a>, <a href="https://www.search.com.vn/wiki/en/Computer-aided_manufacturing" title="Computer-aided manufacturing">computer-aided manufacturing</a> and <a href="https://www.search.com.vn/wiki/en/Optics" title="Optics">optics</a>. Here are examples of other geometric problems whose solution involves solving a quartic equation.</p><p>In <a href="https://www.search.com.vn/wiki/en/Computer-aided_manufacturing" title="Computer-aided manufacturing">computer-aided manufacturing</a>, the torus is a shape that is commonly associated with the <a href="https://www.search.com.vn/wiki/en/Endmill" class="mw-redirect" title="Endmill">endmill</a> cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the <span class="texhtml"><i>z</i></span>-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">[8]</a></sup></p><p>A quartic equation arises also in the process of solving the <a href="https://www.search.com.vn/wiki/en/Crossed_ladders_problem" title="Crossed ladders problem">crossed ladders problem</a>, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">[9]</a></sup></p><p>In optics, <a href="https://www.search.com.vn/wiki/en/Alhazen%27s_problem" title="Alhazen's problem">Alhazen's problem</a> is "<i>Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.</i>" This leads to a quartic equation.<sup id="cite_ref-MacTutor_10-0" class="reference"><a href="#cite_note-MacTutor-10">[10]</a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11">[11]</a></sup><sup id="cite_ref-Weiss_12-0" class="reference"><a href="#cite_note-Weiss-12">[12]</a></sup></p><p>Finding the <a href="https://www.search.com.vn/wiki/en/Distance_of_closest_approach_of_ellipses_and_ellipsoids#Distance_of_closest_approach_of_two_ellipses" class="mw-redirect" title="Distance of closest approach of ellipses and ellipsoids">distance of closest approach of two ellipses</a> involves solving a quartic equation.</p><p>The <a href="https://www.search.com.vn/wiki/en/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a 4×4 <a href="https://www.search.com.vn/wiki/en/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> are the roots of a quartic polynomial which is the <a href="https://www.search.com.vn/wiki/en/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of the matrix.</p><p>The characteristic equation of a fourth-order linear <a href="https://www.search.com.vn/wiki/en/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a> or <a href="https://www.search.com.vn/wiki/en/Differential_equation" title="Differential equation">differential equation</a> is a quartic equation. An example arises in the <a href="https://www.search.com.vn/wiki/en/Bending#Timoshenko-Rayleigh_theory" title="Bending">Timoshenko-Rayleigh theory</a> of beam bending.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13">[13]</a></sup></p><p><a href="https://www.search.com.vn/wiki/en/Intersection_(Euclidean_geometry)" class="mw-redirect" title="Intersection (Euclidean geometry)">Intersections</a> between spheres, cylinders, or other <a href="https://www.search.com.vn/wiki/en/Quadric" title="Quadric">quadrics</a> can be found using quartic equations.</p><h2><span class="mw-headline" id="Inflection_points_and_golden_ratio">Inflection points and golden ratio</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=3" title="Edit section's source code: Inflection points and golden ratio"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><p>Letting <span class="texhtml mvar" style="font-style:italic;">F</span> and <span class="texhtml mvar" style="font-style:italic;">G</span> be the distinct <a href="https://www.search.com.vn/wiki/en/Inflection_point" title="Inflection point">inflection points</a> of the graph of a quartic function, and letting <span class="texhtml mvar" style="font-style:italic;">H</span> be the intersection of the inflection <a href="https://www.search.com.vn/wiki/en/Secant_line" title="Secant line">secant line</a> <span class="texhtml mvar" style="font-style:italic;">FG</span> and the quartic, nearer to <span class="texhtml mvar" style="font-style:italic;">G</span> than to <span class="texhtml mvar" style="font-style:italic;">F</span>, then <span class="texhtml mvar" style="font-style:italic;">G</span> divides <span class="texhtml mvar" style="font-style:italic;">FH</span> into the <a href="https://www.search.com.vn/wiki/en/Golden_section" class="mw-redirect" title="Golden section">golden section</a>:<sup id="cite_ref-14" class="reference"><a href="#cite_note-14">[14]</a></sup></p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {FG}{GH}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \;({\text{the golden ratio}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi>G</mi> </mrow> <mrow> <mi>G</mi> <mi>H</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>φ<!-- φ --></mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>the golden ratio</mtext> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {FG}{GH}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \;({\text{the golden ratio}}).}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a263e031f35ca52684eae607ad026a723d879790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.25ex; height:6.009ex;" alt="{\displaystyle {\frac {FG}{GH}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \;({\text{the golden ratio}}).}"></span></dd></dl><p>Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.</p><h2><span class="mw-headline" id="Solution">Solution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=4" title="Edit section's source code: Solution"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><h3><span class="mw-headline" id="Nature_of_the_roots">Nature of the roots</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=5" title="Edit section's source code: Nature of the roots"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h3><p>Given the general quartic equation</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ccc13a4502686dfb9946be432f7ecaabb03fd90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:29.638ex; height:2.843ex;" alt="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}"></span></dd></dl><p>with real coefficients and <span class="texhtml"><i>a</i> ≠ 0</span> the nature of its roots is mainly determined by the sign of its <a href="https://www.search.com.vn/wiki/en/Discriminant" title="Discriminant">discriminant</a> </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta ={}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}\\&+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e\\&-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mn>256</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>192</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mi>d</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>128</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>144</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> <mo>−<!-- − --></mo> <mn>27</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mn>144</mn> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>6</mn> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> <mo>−<!-- − --></mo> <mn>80</mn> <mi>a</mi> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>e</mi> <mo>+</mo> <mn>18</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>16</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>e</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−<!-- − --></mo> <mn>4</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>27</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>c</mi> <mi>d</mi> <mi>e</mi> <mo>−<!-- − --></mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>e</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta ={}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}\\&+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e\\&-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654adf3cf7f9d0750b278d62a52014c015f6e22a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:62.165ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\Delta ={}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}\\&+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e\\&-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\end{aligned}}}"></span></dd></dl><p>This may be refined by considering the signs of four other polynomials:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=8ac-3b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>8</mn> <mi>a</mi> <mi>c</mi> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=8ac-3b^{2}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f259d8ab915fb9d1e71283e8942079527d646890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.298ex; height:2.843ex;" alt="{\displaystyle P=8ac-3b^{2}}"></span></dd></dl><p>such that <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac"><span class="tion"><span class="num"><i>P</i></span><span class="sr-only">/</span><span class="den">8<i>a</i><sup>2</sup></span></span></span></span> is the second degree coefficient of the associated depressed quartic (see <a href="#Converting_to_a_depressed_quartic">below</a>);</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=b^{3}+8da^{2}-4abc,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>d</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=b^{3}+8da^{2}-4abc,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9d3c15db4590d0026b2d23d842426c205d1bf77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.301ex; height:3.009ex;" alt="{\displaystyle R=b^{3}+8da^{2}-4abc,}"></span></dd></dl><p>such that <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>R</i></span><span class="sr-only">/</span><span class="den">8<i>a</i><sup>3</sup></span></span></span></span> is the first degree coefficient of the associated depressed quartic; </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}=c^{2}-3bd+12ae,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>3</mn> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mn>12</mn> <mi>a</mi> <mi>e</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}=c^{2}-3bd+12ae,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91230d28b95c561bc31b5c6620848a6b73dbaf93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.491ex; height:3.009ex;" alt="{\displaystyle \Delta _{0}=c^{2}-3bd+12ae,}"></span></dd></dl><p>which is 0 if the quartic has a triple root; and</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>64</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>e</mi> <mo>−<!-- − --></mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>16</mn> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> <mo>−<!-- − --></mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mi>d</mi> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7389d0e319e9f087d973c438a059e56d2c9d593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:45.397ex; height:2.843ex;" alt="{\displaystyle D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}}"></span></dd></dl><p>which is 0 if the quartic has two double roots.</p><p>The possible cases for the nature of the roots are as follows:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15">[15]</a></sup></p><ul><li>If <span class="texhtml">∆ < 0</span> then the equation has two distinct real roots and two <a href="https://www.search.com.vn/wiki/en/Complex_conjugate" title="Complex conjugate">complex conjugate</a> non-real roots.</li><li>If <span class="texhtml">∆ > 0</span> then either the equation's four roots are all real or none is.<ul><li>If <span class="texhtml mvar" style="font-style:italic;">P</span> < 0 and <span class="texhtml mvar" style="font-style:italic;">D</span> < 0 then all four roots are real and distinct.</li><li>If <span class="texhtml mvar" style="font-style:italic;">P</span> > 0 or <span class="texhtml mvar" style="font-style:italic;">D</span> > 0 then there are two pairs of non-real complex conjugate roots.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16">[16]</a></sup></li></ul></li><li>If <span class="texhtml">∆ = 0</span> then (and only then) the polynomial has a <a href="https://www.search.com.vn/wiki/en/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiple</a> root. Here are the different cases that can occur:<ul><li>If <span class="texhtml mvar" style="font-style:italic;">P</span> < 0 and <span class="texhtml mvar" style="font-style:italic;">D</span> < 0 and <span class="texhtml">∆<sub>0</sub> ≠ 0</span>, there are a real double root and two real simple roots.</li><li>If <span class="texhtml mvar" style="font-style:italic;">D</span> > 0 or (<span class="texhtml mvar" style="font-style:italic;">P</span> > 0 and (<span class="texhtml mvar" style="font-style:italic;">D</span> ≠ 0 or <span class="texhtml mvar" style="font-style:italic;">R</span> ≠ 0)), there are a real double root and two complex conjugate roots.</li><li>If <span class="texhtml">∆<sub>0</sub> = 0</span> and <span class="texhtml mvar" style="font-style:italic;">D</span> ≠ 0, there are a triple root and a simple root, all real.</li><li>If <span class="texhtml mvar" style="font-style:italic;">D</span> = 0, then:<ul><li>If <span class="texhtml mvar" style="font-style:italic;">P</span> < 0, there are two real double roots.</li><li>If <span class="texhtml mvar" style="font-style:italic;">P</span> > 0 and <span class="texhtml mvar" style="font-style:italic;">R</span> = 0, there are two complex conjugate double roots.</li><li>If <span class="texhtml">∆<sub>0</sub> = 0</span>, all four roots are equal to <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>b</i></span><span class="sr-only">/</span><span class="den">4<i>a</i></span></span></span></span></li></ul></li></ul></li></ul><p>There are some cases that do not seem to be covered, but in fact they cannot occur. For example, <span class="texhtml">∆<sub>0</sub> > 0</span>, <span class="texhtml mvar" style="font-style:italic;">P</span> = 0 and <span class="texhtml mvar" style="font-style:italic;">D</span> ≤ 0 is not one of the cases. In fact, if <span class="texhtml">∆<sub>0</sub> > 0</span> and <span class="texhtml mvar" style="font-style:italic;">P</span> = 0 then <span class="texhtml mvar" style="font-style:italic;">D</span> > 0, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16a^{2}\Delta _{0}=3D+P^{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mi>D</mi> <mo>+</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16a^{2}\Delta _{0}=3D+P^{2};}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf56204b6ef5974557ecbf4da59b423ae825d490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.147ex; height:3.009ex;" alt="{\displaystyle 16a^{2}\Delta _{0}=3D+P^{2};}"></span> so this combination is not possible.</p><h3><span class="mw-headline" id="General_formula_for_roots">General formula for roots</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=6" title="Edit section's source code: General formula for roots"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h3><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="https://www.search.com.vn/wiki/en/File:Quartic_Formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Quartic_Formula.svg/600px-Quartic_Formula.svg.png" decoding="async" width="600" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Quartic_Formula.svg/900px-Quartic_Formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Quartic_Formula.svg/1200px-Quartic_Formula.svg.png 2x" data-file-width="14406" data-file-height="1443" /></a><figcaption>Solution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e875c914963b1d55bab5e508ab829c4837e605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.554ex; height:2.843ex;" alt="{\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0}"></span> written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.</figcaption></figure><p>The four roots <span class="texhtml"><i>x</i><sub>1</sub></span>, <span class="texhtml"><i>x</i><sub>2</sub></span>, <span class="texhtml"><i>x</i><sub>3</sub></span>, and <span class="texhtml"><i>x</i><sub>4</sub></span> for the general quartic equation</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb2377cc4d3c046019c429b2259f7f6a20eee34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:30.025ex; height:2.843ex;" alt="{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,}"></span></dd></dl><p>with <span class="texhtml mvar" style="font-style:italic;">a</span> ≠ 0 are given in the following formula, which is deduced from the one in the section on <a href="#Ferrari's_solution">Ferrari's method</a> by back changing the variables (see <a href="#Converting_to_a_depressed_quartic">§ Converting to a depressed quartic</a>) and using the formulas for the <a href="https://www.search.com.vn/wiki/en/Quadratic_function" title="Quadratic function">quadratic</a> and <a href="https://www.search.com.vn/wiki/en/Cubic_function#General_formula_for_roots" title="Cubic function">cubic equations</a>.</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{1,2}\ &=-{\frac {b}{4a}}-S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p+{\frac {q}{S}}}}\\x_{3,4}\ &=-{\frac {b}{4a}}+S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p-{\frac {q}{S}}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>−<!-- − --></mo> <mi>S</mi> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mn>4</mn> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>S</mi> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>S</mi> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mn>4</mn> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>2</mn> <mi>p</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>S</mi> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1,2}\ &=-{\frac {b}{4a}}-S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p+{\frac {q}{S}}}}\\x_{3,4}\ &=-{\frac {b}{4a}}+S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p-{\frac {q}{S}}}}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7b215b301d88f90349d954ec5eef0bdfa0b0b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:40.527ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}x_{1,2}\ &=-{\frac {b}{4a}}-S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p+{\frac {q}{S}}}}\\x_{3,4}\ &=-{\frac {b}{4a}}+S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p-{\frac {q}{S}}}}\end{aligned}}}"></span></dd></dl><p>where <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span> are the coefficients of the second and of the first degree respectively in the <a href="#Converting_to_a_depressed_quartic">associated depressed quartic</a></p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&={\frac {8ac-3b^{2}}{8a^{2}}}\\q&={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>a</mi> <mi>c</mi> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mn>8</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> </mrow> <mrow> <mn>8</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&={\frac {8ac-3b^{2}}{8a^{2}}}\\q&={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107fed581c12eb5fde59087a6c060ff552936d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:22.647ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}p&={\frac {8ac-3b^{2}}{8a^{2}}}\\q&={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}\end{aligned}}}"></span></dd><dd></dd></dl><p>and where</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S&={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {1}{3a}}\left(Q+{\frac {\Delta _{0}}{Q}}\right)}}\\Q&={\sqrt[{3}]{\frac {\Delta _{1}+{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mtext> </mtext> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Q</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>Q</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mrow> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mn>4</mn> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S&={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {1}{3a}}\left(Q+{\frac {\Delta _{0}}{Q}}\right)}}\\Q&={\sqrt[{3}]{\frac {\Delta _{1}+{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2123cde7bdf9a645323f6ccdc74817db4ddaf3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:33.949ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}S&={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {1}{3a}}\left(Q+{\frac {\Delta _{0}}{Q}}\right)}}\\Q&={\sqrt[{3}]{\frac {\Delta _{1}+{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}}\end{aligned}}}"></span></dd></dl><p>(if <span class="texhtml"><i>S</i> = 0</span> or <span class="texhtml"><i>Q</i> = 0</span>, see <a href="#Special_cases_of_the_formula">§ Special cases of the formula</a>, below)</p><p>with</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta _{0}&=c^{2}-3bd+12ae\\\Delta _{1}&=2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>3</mn> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mn>12</mn> <mi>a</mi> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>9</mn> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>+</mo> <mn>27</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>e</mi> <mo>+</mo> <mn>27</mn> <mi>a</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>72</mn> <mi>a</mi> <mi>c</mi> <mi>e</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta _{0}&=c^{2}-3bd+12ae\\\Delta _{1}&=2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326999db36df4331e71d66679f679a1940fc1b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.74ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\Delta _{0}&=c^{2}-3bd+12ae\\\Delta _{1}&=2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace\end{aligned}}}"></span></dd></dl><p>and</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1}^{2}-4\Delta _{0}^{3}=-27\Delta \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mn>4</mn> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>=</mo> <mo>−<!-- − --></mo> <mn>27</mn> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1}^{2}-4\Delta _{0}^{3}=-27\Delta \ ,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78230d06675f829c8bd30dec3921c9f4b8caec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.378ex; height:3.176ex;" alt="{\displaystyle \Delta _{1}^{2}-4\Delta _{0}^{3}=-27\Delta \ ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> is the aforementioned <a href="https://www.search.com.vn/wiki/en/Discriminant" title="Discriminant">discriminant</a>. For the cube root expression for <i>Q</i>, any of the three cube roots in the complex plane can be used, although if one of them is real that is the natural and simplest one to choose. The mathematical expressions of these last four terms are very similar to those of their <a href="https://www.search.com.vn/wiki/en/Cubic_function#Algebraic_solution" title="Cubic function">cubic counterparts</a>.</dd></dl><h4><span class="mw-headline" id="Special_cases_of_the_formula">Special cases of the formula</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=7" title="Edit section's source code: Special cases of the formula"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta >0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta >0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82c2f0482abb077ecda096056a4f555a68e5751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.844ex; height:2.509ex;" alt="{\displaystyle \Delta >0,}"></span> the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is a non-real complex number. In this case, either all roots are non-real or they are all real. In the latter case, the value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> is also real, despite being expressed in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q;}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de0f853283d353d54608c289337cba6e644fda7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.509ex;" alt="{\displaystyle Q;}"></span> this is <a href="https://www.search.com.vn/wiki/en/Casus_irreducibilis" title="Casus irreducibilis">casus irreducibilis</a> of the cubic function extended to the present context of the quartic. One may prefer to express it in a purely real way, by using <a href="https://www.search.com.vn/wiki/en/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>, as follows:</li></ul><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {2}{3a}}{\sqrt {\Delta _{0}}}\cos {\frac {\varphi }{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mtext> </mtext> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mrow> <mi>cos</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ<!-- φ --></mi> <mn>3</mn> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {2}{3a}}{\sqrt {\Delta _{0}}}\cos {\frac {\varphi }{3}}}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b816774ae3ab2439fada737b6e142f5bf11978b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.101ex; height:6.176ex;" alt="{\displaystyle S={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {2}{3a}}{\sqrt {\Delta _{0}}}\cos {\frac {\varphi }{3}}}}}"></span></dd></dl></dd><dd>where<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arccos \left({\frac {\Delta _{1}}{2{\sqrt {\Delta _{0}^{3}}}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arccos \left({\frac {\Delta _{1}}{2{\sqrt {\Delta _{0}^{3}}}}}\right).}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc40ddaf82bc3983dc07913ae083a17fdbc9dfa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.862ex; height:10.509ex;" alt="{\displaystyle \varphi =\arccos \left({\frac {\Delta _{1}}{2{\sqrt {\Delta _{0}^{3}}}}}\right).}"></span></dd></dl></dd></dl><ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \neq 0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecf280c732124f89f0a9a1095add4807f95e29b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.197ex; height:2.676ex;" alt="{\displaystyle \Delta \neq 0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}=0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb8fd017e48fab19c28142bba2f6be1440226de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.898ex; height:2.509ex;" alt="{\displaystyle \Delta _{0}=0,}"></span> the sign of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}={\sqrt {\Delta _{1}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−<!-- − --></mo> <mn>4</mn> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}={\sqrt {\Delta _{1}^{2}}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16eb01c93310097aa660e9e1439c9ab8deff8e5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.719ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}={\sqrt {\Delta _{1}^{2}}}}"></span> has to be chosen to have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\neq 0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1a1a51a43360be643a37fb21258b3c650c3e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.746ex; height:2.676ex;" alt="{\displaystyle Q\neq 0,}"></span> that is one should define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\Delta _{1}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\Delta _{1}^{2}}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc96ee1d18e88b2c79cd3e09f49643f8f1bc765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.314ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\Delta _{1}^{2}}}}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1},}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f60520c97c82516f97fe10bf09ad4cea8f3b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.637ex; height:2.509ex;" alt="{\displaystyle \Delta _{1},}"></span> maintaining the sign of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6558c2ea284db009b96c672b1a420ecc5465cb98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.637ex; height:2.509ex;" alt="{\displaystyle \Delta _{1}.}"></span></li><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98821c682a9f93bc825f6f15a51500ae96d202b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.407ex; height:2.509ex;" alt="{\displaystyle S=0,}"></span> then one must change the choice of the cube root in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> in order to have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>≠<!-- ≠ --></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\neq 0.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23758e8efdf410a522bb179722b13abc1e30620a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.407ex; height:2.676ex;" alt="{\displaystyle S\neq 0.}"></span> This is always possible except if the quartic may be factored into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x+{\tfrac {b}{4a}}\right)^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mrow> <mn>4</mn> <mi>a</mi> </mrow> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x+{\tfrac {b}{4a}}\right)^{4}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20c048edfa2c3202894c8ca6b3b45b1b8a6139f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.174ex; height:5.176ex;" alt="{\displaystyle \left(x+{\tfrac {b}{4a}}\right)^{4}.}"></span> The result is then correct, but misleading because it hides the fact that no cube root is needed in this case. In fact this case<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="https://www.search.com.vn/wiki/en/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Both kinds or specific? (March 2024)">clarification needed</span></a></i>]</sup> may occur only if the <a href="https://www.search.com.vn/wiki/en/Numerator" class="mw-redirect" title="Numerator">numerator</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> is zero, in which case the associated <a href="#Converting_to_a_depressed_quartic">depressed quartic</a> is biquadratic; it may thus be solved by the method described <a href="#Biquadratic_equation">below</a>.</li><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}=0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb8fd017e48fab19c28142bba2f6be1440226de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.898ex; height:2.509ex;" alt="{\displaystyle \Delta _{0}=0,}"></span> and thus also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{1}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{1}=0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b09fcb73e97d701615af1cc93ecdb3902ff8254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.898ex; height:2.509ex;" alt="{\displaystyle \Delta _{1}=0,}"></span> at least three roots are equal to each other, and the roots are <a href="https://www.search.com.vn/wiki/en/Rational_function" title="Rational function">rational functions</a> of the coefficients. The triple root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> is a common root of the quartic and its second derivative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(6ax^{2}+3bx+c);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mn>6</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(6ax^{2}+3bx+c);}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884e2b37fac1d7d5ccc3b90c677b04506ccda2dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.572ex; height:3.176ex;" alt="{\displaystyle 2(6ax^{2}+3bx+c);}"></span> it is thus also the unique root of the remainder of the <a href="https://www.search.com.vn/wiki/en/Euclidean_division" title="Euclidean division">Euclidean division</a> of the quartic by its second derivative, which is a linear polynomial. The simple root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> can be deduced from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+3x_{0}=-b/a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+3x_{0}=-b/a.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87bf5d31eb7349ddc3edb8176e24926f9004c733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.714ex; height:2.843ex;" alt="{\displaystyle x_{1}+3x_{0}=-b/a.}"></span></li><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{0}\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ<!-- Δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠<!-- ≠ --></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{0}\neq 0,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb121bb27237b7be12e5eb80fd36c36d43a36307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.898ex; height:2.676ex;" alt="{\displaystyle \Delta _{0}\neq 0,}"></span> the above expression for the roots is correct but misleading, hiding the fact that the polynomial is <a href="https://www.search.com.vn/wiki/en/Irreducible_polynomial" title="Irreducible polynomial">reducible</a> and no cube root is needed to represent the roots.</li></ul><h3><span class="mw-headline" id="Simpler_cases">Simpler cases</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=8" title="Edit section's source code: Simpler cases"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h3><h4><span class="mw-headline" id="Reducible_quartics">Reducible quartics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=9" title="Edit section's source code: Reducible quartics"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>Consider the general quartic</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a055778c00d82ec54b83cb94758d948b64a6be82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.986ex; height:3.176ex;" alt="{\displaystyle Q(x)=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.}"></span></dd></dl><p>It is <a href="https://www.search.com.vn/wiki/en/Irreducible_polynomial" title="Irreducible polynomial">reducible</a> if <span class="texhtml"><i>Q</i>(<i>x</i>) = <i>R</i>(<i>x</i>)×<i>S</i>(<i>x</i>)</span>, where <span class="texhtml"><i>R</i>(<i>x</i>)</span> and <span class="texhtml"><i>S</i>(<i>x</i>)</span> are non-constant polynomials with <a href="https://www.search.com.vn/wiki/en/Rational_number" title="Rational number">rational</a> coefficients (or more generally with coefficients in the same <a href="https://www.search.com.vn/wiki/en/Field_(mathematics)" title="Field (mathematics)">field</a> as the coefficients of <span class="texhtml"><i>Q</i>(<i>x</i>)</span>). Such a factorization will take one of two forms:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=(x-x_{1})(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=(x-x_{1})(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0})}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af42a1ca7410eb64c84eb9cf5c6cc4b06f6083f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.074ex; height:3.176ex;" alt="{\displaystyle Q(x)=(x-x_{1})(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0})}"></span></dd></dl><p>or</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=(c_{2}x^{2}+c_{1}x+c_{0})(d_{2}x^{2}+d_{1}x+d_{0}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=(c_{2}x^{2}+c_{1}x+c_{0})(d_{2}x^{2}+d_{1}x+d_{0}).}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7792e366fe74468b0929f37673ad8d1302d517" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.103ex; height:3.176ex;" alt="{\displaystyle Q(x)=(c_{2}x^{2}+c_{1}x+c_{0})(d_{2}x^{2}+d_{1}x+d_{0}).}"></span></dd></dl><p>In either case, the roots of <span class="texhtml"><i>Q</i>(<i>x</i>)</span> are the roots of the factors, which may be computed using the formulas for the roots of a <a href="https://www.search.com.vn/wiki/en/Quadratic_function" title="Quadratic function">quadratic function</a> or <a href="https://www.search.com.vn/wiki/en/Cubic_function" title="Cubic function">cubic function</a>.</p><p>Detecting the existence of such factorizations can be done <a href="https://www.search.com.vn/wiki/en/Resolvent_cubic#Factoring_quartic_polynomials" title="Resolvent cubic">using the resolvent cubic of <span class="texhtml"><i>Q</i>(<i>x</i>)</span></a>. It turns out that:</p><ul><li>if we are working over <span class="texhtml"><b>R</b></span> (that is, if coefficients are restricted to be real numbers) (or, more generally, over some <a href="https://www.search.com.vn/wiki/en/Real_closed_field" title="Real closed field">real closed field</a>) then there is always such a factorization;</li><li>if we are working over <span class="texhtml"><b>Q</b></span> (that is, if coefficients are restricted to be rational numbers) then there is an algorithm to determine whether or not <span class="texhtml"><i>Q</i>(<i>x</i>)</span> is reducible and, if it is, how to express it as a product of polynomials of smaller degree.</li></ul><p>In fact, several methods of solving quartic equations (<a class="mw-selflink-fragment" href="#Ferrari's_solution">Ferrari's method</a>, <a class="mw-selflink-fragment" href="#Descartes'_solution">Descartes' method</a>, and, to a lesser extent, <a class="mw-selflink-fragment" href="#Euler's_solution">Euler's method</a>) are based upon finding such factorizations.</p><h4><span class="mw-headline" id="Biquadratic_equation">Biquadratic equation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=10" title="Edit section's source code: Biquadratic equation"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>If <span class="texhtml"><i>a</i><sub>3</sub> = <i>a</i><sub>1</sub> = 0</span> then the function </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=a_{4}x^{4}+a_{2}x^{2}+a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=a_{4}x^{4}+a_{2}x^{2}+a_{0}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b789a3d820f37d6b0ec51e5ba1b50b37c7314ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.377ex; height:3.176ex;" alt="{\displaystyle Q(x)=a_{4}x^{4}+a_{2}x^{2}+a_{0}}"></span></dd></dl><p>is called a <b>biquadratic function</b>; equating it to zero defines a <b>biquadratic equation</b>, which is easy to solve as follows</p><p>Let the auxiliary variable <span class="texhtml"><i>z</i> = <i>x</i><sup>2</sup></span>.Then <span class="texhtml"><i>Q</i>(<i>x</i>)</span> becomes a <a href="https://www.search.com.vn/wiki/en/Quadratic_function" title="Quadratic function">quadratic</a> <span class="texhtml"><i>q</i></span> in <span class="texhtml"><i>z</i></span>: <span class="texhtml"><i>q</i>(<i>z</i>) = <i>a</i><sub>4</sub><i>z</i><sup>2</sup> + <i>a</i><sub>2</sub><i>z</i> + <i>a</i><sub>0</sub></span>. Let <span class="texhtml"><i>z</i><sub>+</sub></span> and <span class="texhtml"><i>z</i><sub>−</sub></span> be the roots of <span class="texhtml"><i>q</i>(<i>z</i>)</span>. Then the roots of the quartic <span class="texhtml"><i>Q</i>(<i>x</i>)</span> are</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x_{1}&=+{\sqrt {z_{+}}},\\x_{2}&=-{\sqrt {z_{+}}},\\x_{3}&=+{\sqrt {z_{-}}},\\x_{4}&=-{\sqrt {z_{-}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msub> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{1}&=+{\sqrt {z_{+}}},\\x_{2}&=-{\sqrt {z_{+}}},\\x_{3}&=+{\sqrt {z_{-}}},\\x_{4}&=-{\sqrt {z_{-}}}.\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1b4a32dfd6c1b45cc85adb2ad0a50da788e00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:13.217ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}x_{1}&=+{\sqrt {z_{+}}},\\x_{2}&=-{\sqrt {z_{+}}},\\x_{3}&=+{\sqrt {z_{-}}},\\x_{4}&=-{\sqrt {z_{-}}}.\end{aligned}}}"></span></dd></dl><h4><span class="mw-headline" id="Quasi-palindromic_equation">Quasi-palindromic equation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=11" title="Edit section's source code: Quasi-palindromic equation"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>The polynomial</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>m</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67adea069394fc1792da3e807c3247bb1d55be06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.381ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}}"></span></dd></dl><p>is almost <a href="https://www.search.com.vn/wiki/en/Reciprocal_polynomial#Palindromic_polynomial" title="Reciprocal polynomial">palindromic</a>, as <span class="texhtml"><i>P</i>(<i>mx</i>) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>x</i><sup>4</sup></span><span class="sr-only">/</span><span class="den"><i>m</i><sup>2</sup></span></span></span><i>P</i>(<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>m</i></span><span class="sr-only">/</span><span class="den"><i>x</i></span></span></span>)</span> (it is palindromic if <span class="texhtml"><i>m</i> = 1</span>). The change of variables <span class="texhtml"><i>z</i> = <i>x</i> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>m</i></span><span class="sr-only">/</span><span class="den"><i>x</i></span></span></span></span> in <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>P</i>(<i>x</i>)</span><span class="sr-only">/</span><span class="den"><i>x</i><sup>2</sup></span></span></span> = 0</span> produces the <a href="https://www.search.com.vn/wiki/en/Quadratic_equation" title="Quadratic equation">quadratic equation</a> <span class="texhtml"><i>a</i><sub>0</sub><i>z</i><sup>2</sup> + <i>a</i><sub>1</sub><i>z</i> + <i>a</i><sub>2</sub> − 2<i>ma</i><sub>0</sub> = 0</span>. Since <span class="texhtml"><i>x</i><sup>2</sup> − <i>xz</i> + <i>m</i> = 0</span>, the quartic equation <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> may be solved by applying the <a href="https://www.search.com.vn/wiki/en/Quadratic_formula" title="Quadratic formula">quadratic formula</a> twice.</p><h3><span class="mw-headline" id="Solution_methods">Solution methods</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=12" title="Edit section's source code: Solution methods"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h3><h4><span class="mw-headline" id="Converting_to_a_depressed_quartic">Converting to a depressed quartic</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=13" title="Edit section's source code: Converting to a depressed quartic"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>For solving purposes, it is generally better to convert the quartic into a <b>depressed quartic</b> by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.</p><p>Let</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe98f38497243e255aa01d83caa0dbba26a7a28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.524ex; height:3.009ex;" alt="{\displaystyle a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0}"></span></dd></dl><p>be the general quartic equation we want to solve.</p><p>Dividing by <span class="texhtml"><i>a</i><sub>4</sub></span>, provides the equivalent equation <span class="texhtml"><i>x</i><sup>4</sup> + <i>bx</i><sup>3</sup> + <i>cx</i><sup>2</sup> + <i>dx</i> + <i>e</i> = 0</span>, with <span class="texhtml"><i>b</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>a</i><sub>3</sub></span><span class="sr-only">/</span><span class="den"><i>a</i><sub>4</sub></span></span></span></span>, <span class="texhtml"><i>c</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>a</i><sub>2</sub></span><span class="sr-only">/</span><span class="den"><i>a</i><sub>4</sub></span></span></span></span>, <span class="texhtml"><i>d</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>a</i><sub>1</sub></span><span class="sr-only">/</span><span class="den"><i>a</i><sub>4</sub></span></span></span></span>, and <span class="texhtml"><i>e</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>a</i><sub>0</sub></span><span class="sr-only">/</span><span class="den"><i>a</i><sub>4</sub></span></span></span></span>.Substituting <span class="texhtml"><i>y</i> − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>b</i></span><span class="sr-only">/</span><span class="den">4</span></span></span></span> for <span class="texhtml mvar" style="font-style:italic;">x</span> gives, after regrouping the terms, the equation <span class="texhtml"><i>y</i><sup>4</sup> + <i>py</i><sup>2</sup> + <i>qy</i> + <i>r</i> = 0</span>,where</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p&={\frac {8c-3b^{2}}{8}}={\frac {8a_{2}a_{4}-3{a_{3}}^{2}}{8{a_{4}}^{2}}}\\q&={\frac {b^{3}-4bc+8d}{8}}={\frac {{a_{3}}^{3}-4a_{2}a_{3}a_{4}+8a_{1}{a_{4}}^{2}}{8{a_{4}}^{3}}}\\r&={\frac {-3b^{4}+256e-64bd+16b^{2}c}{256}}={\frac {-3{a_{3}}^{4}+256a_{0}{a_{4}}^{3}-64a_{1}a_{3}{a_{4}}^{2}+16a_{2}{a_{3}}^{2}a_{4}}{256{a_{4}}^{4}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>p</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>c</mi> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mn>8</mn> <mi>d</mi> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mn>8</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>256</mn> <mi>e</mi> <mo>−<!-- − --></mo> <mn>64</mn> <mi>b</mi> <mi>d</mi> <mo>+</mo> <mn>16</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mrow> <mn>256</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>256</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>64</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>16</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mn>256</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p&={\frac {8c-3b^{2}}{8}}={\frac {8a_{2}a_{4}-3{a_{3}}^{2}}{8{a_{4}}^{2}}}\\q&={\frac {b^{3}-4bc+8d}{8}}={\frac {{a_{3}}^{3}-4a_{2}a_{3}a_{4}+8a_{1}{a_{4}}^{2}}{8{a_{4}}^{3}}}\\r&={\frac {-3b^{4}+256e-64bd+16b^{2}c}{256}}={\frac {-3{a_{3}}^{4}+256a_{0}{a_{4}}^{3}-64a_{1}a_{3}{a_{4}}^{2}+16a_{2}{a_{3}}^{2}a_{4}}{256{a_{4}}^{4}}}.\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e006e9508b74b68381cdc5a9128ea569c5782d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.672ex; margin-bottom: -0.333ex; width:82.876ex; height:19.009ex;" alt="{\displaystyle {\begin{aligned}p&={\frac {8c-3b^{2}}{8}}={\frac {8a_{2}a_{4}-3{a_{3}}^{2}}{8{a_{4}}^{2}}}\\q&={\frac {b^{3}-4bc+8d}{8}}={\frac {{a_{3}}^{3}-4a_{2}a_{3}a_{4}+8a_{1}{a_{4}}^{2}}{8{a_{4}}^{3}}}\\r&={\frac {-3b^{4}+256e-64bd+16b^{2}c}{256}}={\frac {-3{a_{3}}^{4}+256a_{0}{a_{4}}^{3}-64a_{1}a_{3}{a_{4}}^{2}+16a_{2}{a_{3}}^{2}a_{4}}{256{a_{4}}^{4}}}.\end{aligned}}}"></span></dd></dl><p>If <span class="texhtml"><i>y</i><sub>0</sub></span> is a root of this depressed quartic, then <span class="texhtml"><i>y</i><sub>0</sub> − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>b</i></span><span class="sr-only">/</span><span class="den">4</span></span></span></span> (that is <span class="texhtml"><i>y</i><sub>0</sub> − <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>a</i><sub>3</sub></span><span class="sr-only">/</span><span class="den">4<i>a</i><sub>4</sub></span></span></span>)</span> is a root of the original quartic and every root of the original quartic can be obtained by this process.</p><h4><span id="Ferrari.27s_solution"></span><span class="mw-headline" id="Ferrari's_solution">Ferrari's solution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=14" title="Edit section's source code: Ferrari's solution"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>As explained in the preceding section, we may start with the <i>depressed quartic equation</i></p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{4}+py^{2}+qy+r=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi>y</mi> <mo>+</mo> <mi>r</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{4}+py^{2}+qy+r=0.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0a9ed641cce4096664c58f417a943fb2b730d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.302ex; height:3.009ex;" alt="{\displaystyle y^{4}+py^{2}+qy+r=0.}"></span></dd></dl><p>This depressed quartic can be solved by means of a method discovered by <a href="https://www.search.com.vn/wiki/en/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a>. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(y^{2}+{\frac {p}{2}}\right)^{2}=-qy-r+{\frac {p^{2}}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−<!-- − --></mo> <mi>q</mi> <mi>y</mi> <mo>−<!-- − --></mo> <mi>r</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(y^{2}+{\frac {p}{2}}\right)^{2}=-qy-r+{\frac {p^{2}}{4}}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b2f553b0b2038d5d6c1ed514f9fb9324f1e5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.458ex; height:5.676ex;" alt="{\displaystyle \left(y^{2}+{\frac {p}{2}}\right)^{2}=-qy-r+{\frac {p^{2}}{4}}.}"></span></dd></dl><p>Then, we introduce a variable <span class="texhtml mvar" style="font-style:italic;">m</span> into the factor on the left-hand side by adding <span class="texhtml">2<i>y</i><sup>2</sup><i>m</i> + <i>pm</i> + <i>m</i><sup>2</sup></span> to both sides. After regrouping the coefficients of the power of <span class="texhtml mvar" style="font-style:italic;">y</span> on the right-hand side, this gives the equation </p><table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=2my^{2}-qy+m^{2}+mp+{\frac {p^{2}}{4}}-r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>m</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>q</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>m</mi> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>−<!-- − --></mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=2my^{2}-qy+m^{2}+mp+{\frac {p^{2}}{4}}-r,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde87e38e3fdcc396ba7c126e2862901cdf696ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.774ex; height:5.676ex;" alt="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=2my^{2}-qy+m^{2}+mp+{\frac {p^{2}}{4}}-r,}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table><p>which is equivalent to the original equation, whichever value is given to <span class="texhtml mvar" style="font-style:italic;">m</span>.</p><p>As the value of <span class="texhtml mvar" style="font-style:italic;">m</span> may be arbitrarily chosen, we will choose it in order to <a href="https://www.search.com.vn/wiki/en/Complete_the_square" class="mw-redirect" title="Complete the square">complete the square</a> on the right-hand side. This implies that the <a href="https://www.search.com.vn/wiki/en/Discriminant" title="Discriminant">discriminant</a> in <span class="texhtml mvar" style="font-style:italic;">y</span> of this <a href="https://www.search.com.vn/wiki/en/Quadratic_equation" title="Quadratic equation">quadratic equation</a> is zero, that is <span class="texhtml mvar" style="font-style:italic;">m</span> is a root of the equation</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-q)^{2}-4(2m)\left(m^{2}+pm+{\frac {p^{2}}{4}}-r\right)=0,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>q</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>−<!-- − --></mo> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-q)^{2}-4(2m)\left(m^{2}+pm+{\frac {p^{2}}{4}}-r\right)=0,\,}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2d1f99ed9a8ff005d4cb905a81a91bb1125a5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.793ex; height:6.343ex;" alt="{\displaystyle (-q)^{2}-4(2m)\left(m^{2}+pm+{\frac {p^{2}}{4}}-r\right)=0,\,}"></span></dd></dl><p>which may be rewritten as</p><table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8m^{3}+8pm^{2}+(2p^{2}-8r)m-q^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>p</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>8</mn> <mi>r</mi> <mo stretchy="false">)</mo> <mi>m</mi> <mo>−<!-- − --></mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8m^{3}+8pm^{2}+(2p^{2}-8r)m-q^{2}=0.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d7ef804f15f81814e6feddc600c3bd63f7d7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.534ex; height:3.176ex;" alt="{\displaystyle 8m^{3}+8pm^{2}+(2p^{2}-8r)m-q^{2}=0.}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1a" class="reference nourlexpansion" style="font-weight:bold;">1a</span>)</b></td></tr></tbody></table><p>This is the <a href="https://www.search.com.vn/wiki/en/Resolvent_cubic" title="Resolvent cubic">resolvent cubic</a> of the quartic equation. The value of <span class="texhtml mvar" style="font-style:italic;">m</span> may thus be obtained from <a href="https://www.search.com.vn/wiki/en/Cubic_equation#Cardano's_method" title="Cubic equation">Cardano's formula</a>. When <span class="texhtml mvar" style="font-style:italic;">m</span> is a root of this equation, the right-hand side of equation (<i><b><a href="#math_1">1</a></b></i>) is the square</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <mi>y</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909d8c08516883c9f1fdf52eebb061816a9243f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.394ex; height:6.843ex;" alt="{\displaystyle \left({\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}"></span></dd></dl><p>However, this induces a division by zero if <span class="texhtml"><i>m</i> = 0</span>. This implies <span class="texhtml"><i>q</i> = 0</span>, and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that <span class="texhtml"><i>m</i> ≠ 0</span>. This is always possible except for the depressed equation <span class="texhtml"><i>y</i><sup>4</sup> = 0</span>.</p><p>Now, if <span class="texhtml mvar" style="font-style:italic;">m</span> is a root of the cubic equation such that <span class="texhtml"><i>m</i> ≠ 0</span>, equation (<i><b><a href="#math_1">1</a></b></i>) becomes</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=\left(y{\sqrt {2m}}-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=\left(y{\sqrt {2m}}-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc58d5c3dbf7e1d6047b3bfcf93fa666daab5231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.264ex; height:6.843ex;" alt="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=\left(y{\sqrt {2m}}-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}"></span></dd></dl><p>This equation is of the form <span class="texhtml"><i>M</i><sup>2</sup> = <i>N</i><sup>2</sup></span>, which can be rearranged as <span class="texhtml"><i>M</i><sup>2</sup> − <i>N</i><sup>2</sup> = 0</span> or <span class="texhtml">(<i>M</i> + <i>N</i>)(<i>M</i> − <i>N</i>) = 0</span>. Therefore, equation (<i><b><a href="#math_1">1</a></b></i>) may be rewritten as</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m+{\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)\left(y^{2}+{\frac {p}{2}}+m-{\sqrt {2m}}y+{\frac {q}{2{\sqrt {2m}}}}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <mi>y</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>m</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(y^{2}+{\frac {p}{2}}+m+{\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)\left(y^{2}+{\frac {p}{2}}+m-{\sqrt {2m}}y+{\frac {q}{2{\sqrt {2m}}}}\right)=0.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62755d59466ce3d499d261207bc05db51e2d9bf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:74.245ex; height:6.509ex;" alt="{\displaystyle \left(y^{2}+{\frac {p}{2}}+m+{\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)\left(y^{2}+{\frac {p}{2}}+m-{\sqrt {2m}}y+{\frac {q}{2{\sqrt {2m}}}}\right)=0.}"></span></dd></dl><p>This equation is easily solved by applying to each factor the <a href="https://www.search.com.vn/wiki/en/Quadratic_formula" title="Quadratic formula">quadratic formula</a>. Solving them we may write the four roots as</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2},}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bf0eb43bfbb3d4078e2179186d04355ba49e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.98ex; height:9.009ex;" alt="{\displaystyle y={\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2},}"></span></dd></dl><p>where <span class="texhtml">±<sub>1</sub></span> and <span class="texhtml">±<sub>2</sub></span> denote either <span class="texhtml">+</span> or <span class="texhtml">−</span>. As the two occurrences of <span class="texhtml">±<sub>1</sub></span> must denote the same sign, this leaves four possibilities, one for each root.</p><p>Therefore, the solutions of the original quartic equation are</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-{a_{3} \over 4a_{4}}+{\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> </msqrt> </mrow> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−<!-- − --></mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <msub> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-{a_{3} \over 4a_{4}}+{\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2}.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10c7d843b0a7bf142a14e749ad9f025d8654796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:50.086ex; height:9.343ex;" alt="{\displaystyle x=-{a_{3} \over 4a_{4}}+{\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2}.}"></span></dd></dl><p>A comparison with the <a href="#General_formula_for_roots">general formula</a> above shows that <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2<i>m</i></span></span> = 2<i>S</i></span>.</p><h4><span id="Descartes.27_solution"></span><span class="mw-headline" id="Descartes'_solution">Descartes' solution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=15" title="Edit section's source code: Descartes' solution"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>Descartes<sup id="cite_ref-17" class="reference"><a href="#cite_note-17">[17]</a></sup> introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x^{4}+bx^{3}+cx^{2}+dx+e&=(x^{2}+sx+t)(x^{2}+ux+v)\\&=x^{4}+(s+u)x^{3}+(t+v+su)x^{2}+(sv+tu)x+tv\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> <mo>+</mo> <mi>s</mi> <mi>u</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mi>v</mi> <mo>+</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mi>v</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{4}+bx^{3}+cx^{2}+dx+e&=(x^{2}+sx+t)(x^{2}+ux+v)\\&=x^{4}+(s+u)x^{3}+(t+v+su)x^{2}+(sv+tu)x+tv\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e92268de0b2bb80c179583915f3588de366cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:77.791ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}x^{4}+bx^{3}+cx^{2}+dx+e&=(x^{2}+sx+t)(x^{2}+ux+v)\\&=x^{4}+(s+u)x^{3}+(t+v+su)x^{2}+(sv+tu)x+tv\end{aligned}}}"></span></dd></dl><p>By <a href="https://www.search.com.vn/wiki/en/Equating_coefficients" title="Equating coefficients">equating coefficients</a>, this results in the following system of equations:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}b=s+u\\c=t+v+su\\d=sv+tu\\e=tv\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>b</mi> <mo>=</mo> <mi>s</mi> <mo>+</mo> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> <mo>+</mo> <mi>s</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mo>=</mo> <mi>s</mi> <mi>v</mi> <mo>+</mo> <mi>t</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>e</mi> <mo>=</mo> <mi>t</mi> <mi>v</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}b=s+u\\c=t+v+su\\d=sv+tu\\e=tv\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/112756eef7de0e37a9dba1b605a0deec076431c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:16.991ex; height:12.509ex;" alt="{\displaystyle \left\{{\begin{array}{l}b=s+u\\c=t+v+su\\d=sv+tu\\e=tv\end{array}}\right.}"></span></dd></dl><p>This can be simplified by starting again with the <a href="#Converting_to_a_depressed_quartic">depressed quartic</a> <span class="texhtml"><i>y</i><sup>4</sup> + <i>py</i><sup>2</sup> + <i>qy</i> + <i>r</i></span>, which can be obtained by substituting <span class="texhtml"><i>y</i> − <i>b</i>/4</span> for <span class="texhtml"><i>x</i></span>. Since the coefficient of <span class="texhtml"><i>y</i><sup>3</sup></span> is <span class="texhtml">0</span>, we get <span class="texhtml"><i>s</i> = −<i>u</i></span>, and:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}p+u^{2}=t+v\\q=u(t-v)\\r=tv\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−<!-- − --></mo> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>=</mo> <mi>t</mi> <mi>v</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}p+u^{2}=t+v\\q=u(t-v)\\r=tv\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3519c2a5e2b0b6250a4196bf592cdf6d4cc8ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.117ex; height:9.509ex;" alt="{\displaystyle \left\{{\begin{array}{l}p+u^{2}=t+v\\q=u(t-v)\\r=tv\end{array}}\right.}"></span></dd></dl><p>One can now eliminate both <span class="texhtml mvar" style="font-style:italic;">t</span> and <span class="texhtml mvar" style="font-style:italic;">v</span> by doing the following:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}u^{2}(p+u^{2})^{2}-q^{2}&=u^{2}(t+v)^{2}-u^{2}(t-v)^{2}\\&=u^{2}[(t+v+(t-v))(t+v-(t-v))]\\&=u^{2}(2t)(2v)\\&=4u^{2}tv\\&=4u^{2}r\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−<!-- − --></mo> <mi>v</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−<!-- − --></mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>v</mi> <mo>−<!-- − --></mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−<!-- − --></mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>v</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>t</mi> <mi>v</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}u^{2}(p+u^{2})^{2}-q^{2}&=u^{2}(t+v)^{2}-u^{2}(t-v)^{2}\\&=u^{2}[(t+v+(t-v))(t+v-(t-v))]\\&=u^{2}(2t)(2v)\\&=4u^{2}tv\\&=4u^{2}r\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3872dc7d07870b83c8d97f4fda44076625f066fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:56.291ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}u^{2}(p+u^{2})^{2}-q^{2}&=u^{2}(t+v)^{2}-u^{2}(t-v)^{2}\\&=u^{2}[(t+v+(t-v))(t+v-(t-v))]\\&=u^{2}(2t)(2v)\\&=4u^{2}tv\\&=4u^{2}r\end{aligned}}}"></span></dd></dl><p>If we set <span class="texhtml"><i>U</i> = <i>u</i><sup>2</sup></span>, then solving this equation becomes finding the roots of the <a href="https://www.search.com.vn/wiki/en/Resolvent_cubic" title="Resolvent cubic">resolvent cubic</a></p><table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{3}+2pU^{2}+(p^{2}-4r)U-q^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>p</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>r</mi> <mo stretchy="false">)</mo> <mi>U</mi> <mo>−<!-- − --></mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{3}+2pU^{2}+(p^{2}-4r)U-q^{2},}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3100a7e606f67f2e9573d2fda4194993599cd4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.292ex; height:3.176ex;" alt="{\displaystyle U^{3}+2pU^{2}+(p^{2}-4r)U-q^{2},}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table><p>which is <a href="https://www.search.com.vn/wiki/en/Cubic_function#General_solution_to_the_cubic_equation_with_real_coefficients" title="Cubic function">done elsewhere</a>. This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m.</p><p>If <span class="texhtml"><i>u</i></span> is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic <span class="texhtml"><i>x</i><sup>4</sup></span>, which is trivially factored),</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}s=-u\\2t=p+u^{2}+q/u\\2v=p+u^{2}-q/u\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>s</mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>t</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>v</mi> <mo>=</mo> <mi>p</mi> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>u</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}s=-u\\2t=p+u^{2}+q/u\\2v=p+u^{2}-q/u\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5596883a6b4740f1f72a064c413168cf1f36b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:21.002ex; height:9.843ex;" alt="{\displaystyle \left\{{\begin{array}{l}s=-u\\2t=p+u^{2}+q/u\\2v=p+u^{2}-q/u\end{array}}\right.}"></span></dd></dl><p>The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of <span class="texhtml mvar" style="font-style:italic;">u</span> for the square root of <span class="texhtml mvar" style="font-style:italic;">U</span> merely exchanges the two quadratics with one another.</p><p>The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic (<i><b><a href="#math_2">2</a></b></i>) has a non-zero root which is the square of a rational, or <span class="texhtml"><i>p</i><sup>2</sup> − 4<i>r</i></span> is the square of rational and <span class="texhtml"><i>q</i> = 0</span>; this can readily be checked using the <a href="https://www.search.com.vn/wiki/en/Rational_root_test" class="mw-redirect" title="Rational root test">rational root test</a>.<sup id="cite_ref-Brookfield_18-0" class="reference"><a href="#cite_note-Brookfield-18">[18]</a></sup></p><h4><span id="Euler.27s_solution"></span><span class="mw-headline" id="Euler's_solution">Euler's solution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=16" title="Edit section's source code: Euler's solution"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>A variant of the previous method is due to <a href="https://www.search.com.vn/wiki/en/Leonhard_Euler" title="Leonhard Euler">Euler</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19">[19]</a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20">[20]</a></sup> Unlike the previous methods, both of which use <i>some</i> root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic <span class="texhtml"><i>x</i><sup>4</sup> + <i>px</i><sup>2</sup> + <i>qx</i> + <i>r</i></span>. Observe that, if</p><ul><li><span class="texhtml"><i>x</i><sup>4</sup> + <i>px</i><sup>2</sup> + <i>qx</i> + <i>r</i> = (<i>x</i><sup>2</sup> + <i>sx</i> + <i>t</i>)(<i>x</i><sup>2</sup> − <i>sx</i> + <i>v</i>)</span>,</li><li><span class="texhtml"><i>r</i><sub>1</sub></span> and <span class="texhtml"><i>r</i><sub>2</sub></span> are the roots of <span class="texhtml"><i>x</i><sup>2</sup> + <i>sx</i> + <i>t</i></span>,</li><li><span class="texhtml"><i>r</i><sub>3</sub></span> and <span class="texhtml"><i>r</i><sub>4</sub></span> are the roots of <span class="texhtml"><i>x</i><sup>2</sup> − <i>sx</i> + <i>v</i></span>,</li></ul><p>then</p><ul><li>the roots of <span class="texhtml"><i>x</i><sup>4</sup> + <i>px</i><sup>2</sup> + <i>qx</i> + <i>r</i></span> are <span class="texhtml"><i>r</i><sub>1</sub></span>, <span class="texhtml"><i>r</i><sub>2</sub></span>, <span class="texhtml"><i>r</i><sub>3</sub></span>, and <span class="texhtml"><i>r</i><sub>4</sub></span>,</li><li><span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> = −<i>s</i></span>,</li><li><span class="texhtml"><i>r</i><sub>3</sub> + <i>r</i><sub>4</sub> = <i>s</i></span>.</li></ul><p>Therefore, <span class="texhtml">(<i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>)(<i>r</i><sub>3</sub> + <i>r</i><sub>4</sub>) = −<i>s</i><sup>2</sup></span>. In other words, <span class="texhtml">−(<i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>)(<i>r</i><sub>3</sub> + <i>r</i><sub>4</sub>)</span> is one of the roots of the resolvent cubic (<i><b><a href="#math_2">2</a></b></i>) and this suggests that the roots of that cubic are equal to <span class="texhtml">−(<i>r</i><sub>1</sub> + <i>r</i><sub>2</sub>)(<i>r</i><sub>3</sub> + <i>r</i><sub>4</sub>)</span>, <span class="texhtml">−(<i>r</i><sub>1</sub> + <i>r</i><sub>3</sub>)(<i>r</i><sub>2</sub> + <i>r</i><sub>4</sub>)</span>, and <span class="texhtml">−(<i>r</i><sub>1</sub> + <i>r</i><sub>4</sub>)(<i>r</i><sub>2</sub> + <i>r</i><sub>3</sub>)</span>. This is indeed true and it follows from <a href="https://www.search.com.vn/wiki/en/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formulas</a>. It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that <span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> + <i>r</i><sub>3</sub> + <i>r</i><sub>4</sub> = 0</span>. (Of course, this also follows from the fact that <span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> + <i>r</i><sub>3</sub> + <i>r</i><sub>4</sub> = −<i>s</i> + <i>s</i></span>.) Therefore, if <span class="texhtml"><i>α</i></span>, <span class="texhtml"><i>β</i></span>, and <span class="texhtml"><i>γ</i></span> are the roots of the resolvent cubic, then the numbers <span class="texhtml"><i>r</i><sub>1</sub></span>, <span class="texhtml"><i>r</i><sub>2</sub></span>, <span class="texhtml"><i>r</i><sub>3</sub></span>, and <span class="texhtml"><i>r</i><sub>4</sub></span> are such that</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\(r_{1}+r_{2})(r_{3}+r_{4})=-\alpha \\(r_{1}+r_{3})(r_{2}+r_{4})=-\beta \\(r_{1}+r_{4})(r_{2}+r_{3})=-\gamma {\text{.}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mi>α<!-- α --></mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mi>γ<!-- γ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\(r_{1}+r_{2})(r_{3}+r_{4})=-\alpha \\(r_{1}+r_{3})(r_{2}+r_{4})=-\beta \\(r_{1}+r_{4})(r_{2}+r_{3})=-\gamma {\text{.}}\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23603efbc9071ee51e52121c995c352c09a90ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:27.344ex; height:12.843ex;" alt="{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\(r_{1}+r_{2})(r_{3}+r_{4})=-\alpha \\(r_{1}+r_{3})(r_{2}+r_{4})=-\beta \\(r_{1}+r_{4})(r_{2}+r_{3})=-\gamma {\text{.}}\end{array}}\right.}"></span></dd></dl><p>It is a consequence of the first two equations that <span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub></span> is a square root of <span class="texhtml"><i>α</i></span> and that <span class="texhtml"><i>r</i><sub>3</sub> + <i>r</i><sub>4</sub></span> is the other square root of <span class="texhtml"><i>α</i></span>. For the same reason,</p><ul><li><span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>3</sub></span> is a square root of <span class="texhtml"><i>β</i></span>,</li><li><span class="texhtml"><i>r</i><sub>2</sub> + <i>r</i><sub>4</sub></span> is the other square root of <span class="texhtml"><i>β</i></span>,</li><li><span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>4</sub></span> is a square root of <span class="texhtml"><i>γ</i></span>,</li><li><span class="texhtml"><i>r</i><sub>2</sub> + <i>r</i><sub>3</sub></span> is the other square root of <span class="texhtml"><i>γ</i></span>.</li></ul><p>Therefore, the numbers <span class="texhtml"><i>r</i><sub>1</sub></span>, <span class="texhtml"><i>r</i><sub>2</sub></span>, <span class="texhtml"><i>r</i><sub>3</sub></span>, and <span class="texhtml"><i>r</i><sub>4</sub></span> are such that</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\r_{1}+r_{2}={\sqrt {\alpha }}\\r_{1}+r_{3}={\sqrt {\beta }}\\r_{1}+r_{4}={\sqrt {\gamma }}{\text{;}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ<!-- γ --></mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>;</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\r_{1}+r_{2}={\sqrt {\alpha }}\\r_{1}+r_{3}={\sqrt {\beta }}\\r_{1}+r_{4}={\sqrt {\gamma }}{\text{;}}\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b633028854053acce7947078b30e25a8d7d9197c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:24.011ex; height:13.843ex;" alt="{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\r_{1}+r_{2}={\sqrt {\alpha }}\\r_{1}+r_{3}={\sqrt {\beta }}\\r_{1}+r_{4}={\sqrt {\gamma }}{\text{;}}\end{array}}\right.}"></span></dd></dl><p>the sign of the square roots will be dealt with below. The only solution of this system is:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{array}{l}r_{1}={\frac {{\sqrt {\alpha }}+{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}\\[2mm]r_{2}={\frac {{\sqrt {\alpha }}-{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{3}={\frac {-{\sqrt {\alpha }}+{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{4}={\frac {-{\sqrt {\alpha }}-{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}{\text{.}}\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left" rowspacing="0.967em 0.967em 0.967em 0.4em" columnspacing="1em"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ<!-- γ --></mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ<!-- γ --></mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ<!-- γ --></mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ<!-- γ --></mi> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{array}{l}r_{1}={\frac {{\sqrt {\alpha }}+{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}\\[2mm]r_{2}={\frac {{\sqrt {\alpha }}-{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{3}={\frac {-{\sqrt {\alpha }}+{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{4}={\frac {-{\sqrt {\alpha }}-{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}{\text{.}}\end{array}}\right.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f811e5a1fae80ee4814a27578d9ebf90528a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:20.605ex; height:23.843ex;" alt="{\displaystyle \left\{{\begin{array}{l}r_{1}={\frac {{\sqrt {\alpha }}+{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}\\[2mm]r_{2}={\frac {{\sqrt {\alpha }}-{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{3}={\frac {-{\sqrt {\alpha }}+{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\[2mm]r_{4}={\frac {-{\sqrt {\alpha }}-{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}{\text{.}}\end{array}}\right.}"></span></dd></dl><p>Since, in general, there are two choices for each square root, it might look as if this provides <span class="texhtml">8 (= 2<sup>3</sup>)</span> choices for the set <span class="texhtml">{<i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub></span>}, but, in fact, it provides no more than <span class="texhtml">2</span> such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set <span class="texhtml">{<i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, <i>r</i><sub>3</sub>, <i>r</i><sub>4</sub></span>} becomes the set <span class="texhtml">{−<i>r</i><sub>1</sub>, −<i>r</i><sub>2</sub>, −<i>r</i><sub>3</sub>, −<i>r</i><sub>4</sub></span>}.</p><p>In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers <span class="texhtml"><i>α</i></span>, <span class="texhtml"><i>β</i></span>, and <span class="texhtml"><i>γ</i></span> and uses them to compute the numbers <span class="texhtml"><i>r</i><sub>1</sub></span>, <span class="texhtml"><i>r</i><sub>2</sub></span>, <span class="texhtml"><i>r</i><sub>3</sub></span>, and <span class="texhtml"><i>r</i><sub>4</sub></span> from the previous equalities. Then, one computes the number <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>α</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>β</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>γ</i></span></span></span>. Since <span class="texhtml"><i>α</i></span>, <span class="texhtml"><i>β</i></span>, and <span class="texhtml"><i>γ</i></span> are the roots of (<i><b><a href="#math_2">2</a></b></i>), it is a consequence of Vieta's formulas that their product is equal to <span class="texhtml"><i>q</i><sup>2</sup></span> and therefore that <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>α</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>β</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>γ</i></span></span> = ±<i>q</i></span>. But a straightforward computation shows that</p><dl><dd><span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>α</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>β</i></span></span><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>γ</i></span></span> = <i>r</i><sub>1</sub><i>r</i><sub>2</sub><i>r</i><sub>3</sub> + <i>r</i><sub>1</sub><i>r</i><sub>2</sub><i>r</i><sub>4</sub> + <i>r</i><sub>1</sub><i>r</i><sub>3</sub><i>r</i><sub>4</sub> + <i>r</i><sub>2</sub><i>r</i><sub>3</sub><i>r</i><sub>4</sub>.</span></dd></dl><p>If this number is <span class="texhtml">−<i>q</i></span>, then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be <span class="texhtml">−<i>r</i><sub>1</sub></span>, <span class="texhtml">−<i>r</i><sub>2</sub></span>, <span class="texhtml">−<i>r</i><sub>3</sub></span>, and <span class="texhtml">−<i>r</i><sub>4</sub></span>, which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one).</p><p>This argument suggests another way of choosing the square roots:</p><ul><li>pick <i>any</i> square root <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>α</i></span></span></span> of <span class="texhtml"><i>α</i></span> and <i>any</i> square root <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>β</i></span></span></span> of <span class="texhtml"><i>β</i></span>;</li><li><i>define</i> <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>γ</i></span></span></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {q}{{\sqrt {\alpha }}{\sqrt {\beta }}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α<!-- α --></mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>β<!-- β --></mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {q}{{\sqrt {\alpha }}{\sqrt {\beta }}}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8f63d9fee86d04592c9a08f26950e5979bec7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.724ex; height:6.176ex;" alt="{\displaystyle -{\frac {q}{{\sqrt {\alpha }}{\sqrt {\beta }}}}}"></span>.</li></ul><p>Of course, this will make no sense if <span class="texhtml"><i>α</i></span> or <span class="texhtml"><i>β</i></span> is equal to <span class="texhtml">0</span>, but <span class="texhtml">0</span> is a root of (<i><b><a href="#math_2">2</a></b></i>) only when <span class="texhtml"><i>q</i> = 0</span>, that is, only when we are dealing with a <a class="mw-selflink-fragment" href="#Biquadratic_equation">biquadratic equation</a>, in which case there is a much simpler approach.</p><h4><span class="mw-headline" id="Solving_by_Lagrange_resolvent">Solving by Lagrange resolvent</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=17" title="Edit section's source code: Solving by Lagrange resolvent"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>The <a href="https://www.search.com.vn/wiki/en/Symmetric_group" title="Symmetric group">symmetric group</a> <span class="texhtml"><i>S</i><sub>4</sub></span> on four elements has the <a href="https://www.search.com.vn/wiki/en/Klein_four-group" title="Klein four-group">Klein four-group</a> as a <a href="https://www.search.com.vn/wiki/en/Normal_subgroup" title="Normal subgroup">normal subgroup</a>. This suggests using a <b><style data-mw-deduplicate="TemplateStyles:r1023754711">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}</style><span class="vanchor"><span id="resolvent_cubic"></span><span class="vanchor-text">resolvent cubic</span></span></b> whose roots may be variously described as a discrete Fourier transform or a <a href="https://www.search.com.vn/wiki/en/Hadamard_matrix" title="Hadamard matrix">Hadamard matrix</a> transform of the roots; see <a href="https://www.search.com.vn/wiki/en/Lagrange_resolvents" class="mw-redirect" title="Lagrange resolvents">Lagrange resolvents</a> for the general method. Denote by <span class="texhtml"><i>x<sub>i</sub></i></span>, for <span class="texhtml"><i>i</i></span> from <span class="texhtml">0</span> to <span class="texhtml">3</span>, the four roots of <span class="texhtml"><i>x</i><sup>4</sup> + <i>bx</i><sup>3</sup> + <i>cx</i><sup>2</sup> + <i>dx</i> + <i>e</i></span>. If we set</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{0}&={\tfrac {1}{2}}(x_{0}+x_{1}+x_{2}+x_{3}),\\[4pt]s_{1}&={\tfrac {1}{2}}(x_{0}-x_{1}+x_{2}-x_{3}),\\[4pt]s_{2}&={\tfrac {1}{2}}(x_{0}+x_{1}-x_{2}-x_{3}),\\[4pt]s_{3}&={\tfrac {1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−<!-- − --></mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{0}&={\tfrac {1}{2}}(x_{0}+x_{1}+x_{2}+x_{3}),\\[4pt]s_{1}&={\tfrac {1}{2}}(x_{0}-x_{1}+x_{2}-x_{3}),\\[4pt]s_{2}&={\tfrac {1}{2}}(x_{0}+x_{1}-x_{2}-x_{3}),\\[4pt]s_{3}&={\tfrac {1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b31fb6bba5dc3d511b8fdf04a9581d454224f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:28.166ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}s_{0}&={\tfrac {1}{2}}(x_{0}+x_{1}+x_{2}+x_{3}),\\[4pt]s_{1}&={\tfrac {1}{2}}(x_{0}-x_{1}+x_{2}-x_{3}),\\[4pt]s_{2}&={\tfrac {1}{2}}(x_{0}+x_{1}-x_{2}-x_{3}),\\[4pt]s_{3}&={\tfrac {1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}}"></span></dd></dl><p>then since the transformation is an <a href="https://www.search.com.vn/wiki/en/Involution_(mathematics)" title="Involution (mathematics)">involution</a> we may express the roots in terms of the four <span class="texhtml"><i>s<sub>i</sub></i></span> in exactly the same way. Since we know the value <span class="texhtml"><i>s</i><sub>0</sub> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num"><i>b</i></span><span class="sr-only">/</span><span class="den">2</span></span></span></span>, we only need the values for <span class="texhtml"><i>s</i><sub>1</sub></span>, <span class="texhtml"><i>s</i><sub>2</sub></span> and <span class="texhtml"><i>s</i><sub>3</sub></span>. These are the roots of the polynomial</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{2}-{s_{1}}^{2})(s^{2}-{s_{2}}^{2})(s^{2}-{s_{3}}^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s^{2}-{s_{1}}^{2})(s^{2}-{s_{2}}^{2})(s^{2}-{s_{3}}^{2}).}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9914674acca6a625c6a58f1aa89f0345ded854a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.627ex; height:3.176ex;" alt="{\displaystyle (s^{2}-{s_{1}}^{2})(s^{2}-{s_{2}}^{2})(s^{2}-{s_{3}}^{2}).}"></span></dd></dl><p>Substituting the <span class="texhtml"><i>s<sub>i</sub></i></span> by their values in term of the <span class="texhtml"><i>x<sub>i</sub></i></span>, this polynomial may be expanded in a polynomial in <span class="texhtml"><i>s</i></span> whose coefficients are <a href="https://www.search.com.vn/wiki/en/Symmetric_polynomial" title="Symmetric polynomial">symmetric polynomials</a> in the <span class="texhtml"><i>x<sub>i</sub></i></span>. By the <a href="https://www.search.com.vn/wiki/en/Fundamental_theorem_of_symmetric_polynomials" class="mw-redirect" title="Fundamental theorem of symmetric polynomials">fundamental theorem of symmetric polynomials</a>, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is <span class="texhtml"><i>b</i> = 0</span>, this results in the polynomial</p><table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{6}+2cs^{4}+(c^{2}-4e)s^{2}-d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>c</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mn>4</mn> <mi>e</mi> <mo stretchy="false">)</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{6}+2cs^{4}+(c^{2}-4e)s^{2}-d^{2}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752708315c0d4b320991e74aaefed43a52e144c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.354ex; height:3.176ex;" alt="{\displaystyle s^{6}+2cs^{4}+(c^{2}-4e)s^{2}-d^{2}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table><p>This polynomial is of degree six, but only of degree three in <span class="texhtml"><i>s</i><sup>2</sup></span>, and so the corresponding equation is solvable by the method described in the article about <a href="https://www.search.com.vn/wiki/en/Cubic_function" title="Cubic function">cubic function</a>. By substituting the roots in the expression of the <span class="texhtml"><i>x<sub>i</sub></i></span> in terms of the <span class="texhtml"><i>s<sub>i</sub></i></span>, we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the <span class="texhtml"><i>x<sub>i</sub></i></span>.</p><p>These expressions are unnecessarily complicated, involving the <a href="https://www.search.com.vn/wiki/en/Root_of_unity" title="Root of unity">cubic roots of unity</a>, which can be avoided as follows. If <span class="texhtml"><i>s</i></span> is any non-zero root of (<i><b><a href="#math_3">3</a></b></i>), and if we set</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{1}(x)&=x^{2}+sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}-{\frac {d}{2s}}\\F_{2}(x)&=x^{2}-sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}+{\frac {d}{2s}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>s</mi> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{1}(x)&=x^{2}+sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}-{\frac {d}{2s}}\\F_{2}(x)&=x^{2}-sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}+{\frac {d}{2s}}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c21ce2d30a1a63b92fcacc8eb6f32853c9d2340f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:33.772ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}F_{1}(x)&=x^{2}+sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}-{\frac {d}{2s}}\\F_{2}(x)&=x^{2}-sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}+{\frac {d}{2s}}\end{aligned}}}"></span></dd></dl><p>then</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{1}(x)\times F_{2}(x)=x^{4}+cx^{2}+dx+e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>×<!-- × --></mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}(x)\times F_{2}(x)=x^{4}+cx^{2}+dx+e.}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d745311e7582bcf5cf97ef377ace57e171b97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.886ex; height:3.176ex;" alt="{\displaystyle F_{1}(x)\times F_{2}(x)=x^{4}+cx^{2}+dx+e.}"></span></dd></dl><p>We therefore can solve the quartic by solving for <span class="texhtml"><i>s</i></span> and then solving for the roots of the two factors using the <a href="https://www.search.com.vn/wiki/en/Quadratic_formula" title="Quadratic formula">quadratic formula</a>.</p><p>This gives exactly the same formula for the roots as the one provided by <a class="mw-selflink-fragment" href="#Descartes'_solution">Descartes' method</a>.</p><h4><span class="mw-headline" id="Solving_with_algebraic_geometry">Solving with algebraic geometry</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=18" title="Edit section's source code: Solving with algebraic geometry"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h4><p>There is an alternative solution using algebraic geometry<sup id="cite_ref-21" class="reference"><a href="#cite_note-21">[21]</a></sup> In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three <a href="https://www.search.com.vn/wiki/en/Degenerate_conic" title="Degenerate conic">reducible quadratic curves</a> (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic.</p><p>The four roots of the depressed quartic <span class="texhtml"><i>x</i><sup>4</sup> + <i>px</i><sup>2</sup> + <i>qx</i> + <i>r</i> = 0</span> may also be expressed as the <span class="texhtml mvar" style="font-style:italic;">x</span> coordinates of the intersections of the two quadratic equations <span class="texhtml"><i>y</i><sup>2</sup> + <i>py</i> + <i>qx</i> + <i>r</i> = 0</span> and <span class="texhtml"><i>y</i> − <i>x</i><sup>2</sup> = 0</span> i.e., using the substitution <span class="texhtml"><i>y</i> = <i>x</i><sup>2</sup></span> that two quadratics intersect in four points is an instance of <a href="https://www.search.com.vn/wiki/en/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a>. Explicitly, the four points are <span class="texhtml"><i>P<sub>i</sub></i> ≔ (<i>x<sub>i</sub></i>, <i>x<sub>i</sub></i><sup>2</sup>)</span> for the four roots <span class="texhtml"><i>x<sub>i</sub></i></span> of the quartic.</p><p>These four points are not collinear because they lie on the irreducible quadratic <span class="texhtml"><i>y</i> = <i>x</i><sup>2</sup></span> and thus there is a 1-parameter family of quadratics (a <a href="https://www.search.com.vn/wiki/en/Pencil_of_curves" class="mw-redirect" title="Pencil of curves">pencil of curves</a>) passing through these points. Writing the projectivization of the two quadratics as <a href="https://www.search.com.vn/wiki/en/Quadratic_form" title="Quadratic form">quadratic forms</a> in three variables:</p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}F_{1}(X,Y,Z)&:=Y^{2}+pYZ+qXZ+rZ^{2},\\F_{2}(X,Y,Z)&:=YZ-X^{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>:=</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>Y</mi> <mi>Z</mi> <mo>+</mo> <mi>q</mi> <mi>X</mi> <mi>Z</mi> <mo>+</mo> <mi>r</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>:=</mo> <mi>Y</mi> <mi>Z</mi> <mo>−<!-- − --></mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F_{1}(X,Y,Z)&:=Y^{2}+pYZ+qXZ+rZ^{2},\\F_{2}(X,Y,Z)&:=YZ-X^{2}\end{aligned}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf66021fa062d566910a65318e39adaf550912c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:41.643ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}F_{1}(X,Y,Z)&:=Y^{2}+pYZ+qXZ+rZ^{2},\\F_{2}(X,Y,Z)&:=YZ-X^{2}\end{aligned}}}"></span></dd></dl><p>the pencil is given by the forms <span class="texhtml"><i>λF</i><sub>1</sub> + <i>μF</i><sub>2</sub></span> for any point <span class="texhtml">[<i>λ</i>, <i>μ</i>]</span> in the projective line — in other words, where <span class="texhtml"><i>λ</i></span> and <span class="texhtml"><i>μ</i></span> are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros.</p><p>This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\binom {4}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mn>4</mn> <mn>2</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\binom {4}{2}}}</annotation> </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1867fc0d32b423866c00be3ad31c94eebbecf2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.952ex; height:3.343ex;" alt="{\displaystyle \textstyle {\binom {4}{2}}}"></span> = <span class="texhtml">6</span> different ways. Denote these <span class="texhtml"><i>Q</i><sub>1</sub> = <i>L</i><sub>12</sub> + <i>L</i><sub>34</sub></span>, <span class="texhtml"><i>Q</i><sub>2</sub> = <i>L</i><sub>13</sub> + <i>L</i><sub>24</sub></span>, and <span class="texhtml"><i>Q</i><sub>3</sub> = <i>L</i><sub>14</sub> + <i>L</i><sub>23</sub></span>. Given any two of these, their intersection has exactly the four points.</p><p>The reducible quadratics, in turn, may be determined by expressing the quadratic form <span class="texhtml"><i>λF</i><sub>1</sub> + <i>μF</i><sub>2</sub></span> as a <span class="texhtml">3×3</span> matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in <span class="texhtml"><i>λ</i></span> and <span class="texhtml"><i>μ</i></span> and corresponds to the resolvent cubic.</p><h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=19" title="Edit section's source code: See also"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><ul><li><a href="https://www.search.com.vn/wiki/en/Linear_function" title="Linear function">Linear function</a> – Linear map or polynomial function of degree one</li><li><a href="https://www.search.com.vn/wiki/en/Quadratic_function" title="Quadratic function">Quadratic function</a> – Polynomial function of degree two</li><li><a href="https://www.search.com.vn/wiki/en/Cubic_function" title="Cubic function">Cubic function</a> – Polynomial function of degree 3</li><li><a href="https://www.search.com.vn/wiki/en/Quintic_function" title="Quintic function">Quintic function</a> – Polynomial function of degree 5</li></ul><h2><span class="mw-headline" id="Notes">Notes</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=20" title="Edit section's source code: Notes"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><dl><dd><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation wikicite" id="endnote_Alpha"><b><a href="#ref_Alpha">^α</a></b></span> For the purposes of this article, <i>e</i> is used as a <a href="https://www.search.com.vn/wiki/en/Variable_(mathematics)" title="Variable (mathematics)">variable</a> as opposed to its conventional use as <a href="https://www.search.com.vn/wiki/en/E_(mathematical_constant)" title="E (mathematical constant)">Euler's number</a>(except when otherwise specified).</dd></dl><h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=21" title="Edit section's source code: References"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><style data-mw-deduplicate="TemplateStyles:r1217336898">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"><div class="mw-references-wrap mw-references-columns"><ol class="references"><li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1215172403">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a{background-size:contain}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a{background-size:contain}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a{background-size:contain}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#2C882D;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911F}html.skin-theme-clientpref-night .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-night .mw-parser-output .cs1-hidden-error{color:#f8a397}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-os .mw-parser-output .cs1-hidden-error{color:#f8a397}html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911F}}</style><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="https://www.search.com.vn/wiki/en/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Ferrari.html">"Lodovico Ferrari"</a>, <i><a href="https://www.search.com.vn/wiki/en/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="https://www.search.com.vn/wiki/en/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lodovico+Ferrari&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FFerrari.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFCardano1993" class="citation cs2"><a href="https://www.search.com.vn/wiki/en/Gerolamo_Cardano" title="Gerolamo Cardano">Cardano, Gerolamo</a> (1993) [1545], <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/arsmagnaorruleso0000card"><i>Ars magna or The Rules of Algebra</i></a></span>, Dover, <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/0-486-67811-3" title="Special:BookSources/0-486-67811-3"><bdi>0-486-67811-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ars+magna+or+The+Rules+of+Algebra&rft.pub=Dover&rft.date=1993&rft.isbn=0-486-67811-3&rft.aulast=Cardano&rft.aufirst=Gerolamo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Farsmagnaorruleso0000card&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFDepman1954" class="citation cs2 cs1-prop-foreign-lang-source">Depman (1954), <i>Rasskazy o matematike</i> (in Russian), Leningrad: Gosdetizdat</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rasskazy+o+matematike&rft.place=Leningrad&rft.pub=Gosdetizdat&rft.date=1954&rft.au=Depman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFP._Beckmann1971" class="citation book cs1">P. Beckmann (1971). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80"><i>A history of π</i></a>. Macmillan. p. 80. <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/9780312381851" title="Special:BookSources/9780312381851"><bdi>9780312381851</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+%CF%80&rft.pages=80&rft.pub=Macmillan&rft.date=1971&rft.isbn=9780312381851&rft.au=P.+Beckmann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTB6jzz3ZDTEC%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFP._Beckmann1971" class="citation book cs1">P. Beckmann (1971). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TB6jzz3ZDTEC&pg=PA80"><i>A history of π</i></a>. Macmillan. p. 191. <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/9780312381851" title="Special:BookSources/9780312381851"><bdi>9780312381851</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+%CF%80&rft.pages=191&rft.pub=Macmillan&rft.date=1971&rft.isbn=9780312381851&rft.au=P.+Beckmann&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTB6jzz3ZDTEC%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFP._Zoll1989" class="citation journal cs1">P. Zoll (1989). "Letter to the Editor". <i>American Mathematical Monthly</i>. <b>96</b> (8): 709–710. <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2324719">2324719</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Letter+to+the+Editor&rft.volume=96&rft.issue=8&rft.pages=709-710&rft.date=1989&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324719%23id-name%3DJSTOR&rft.au=P.+Zoll&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Stewart, Ian, <i>Galois Theory, Third Edition</i> (Chapman & Hall/CRC Mathematics, 2004)</span></li><li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://people.math.gatech.edu/~etnyre/class/4441Fall16/ShifrinDiffGeo.pdf">"DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces, p. 36"</a> <span class="cs1-format">(PDF)</span>. <i>math.gatech.edu</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.gatech.edu&rft.atitle=DIFFERENTIAL+GEOMETRY%3A+A+First+Course+in+Curves+and+Surfaces%2C+p.+36&rft_id=http%3A%2F%2Fpeople.math.gatech.edu%2F~etnyre%2Fclass%2F4441Fall16%2FShifrinDiffGeo.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CrossedLaddersProblem.html">"Crossed Ladders Problem"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Crossed+Ladders+Problem&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FCrossedLaddersProblem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-MacTutor-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-MacTutor_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="https://www.search.com.vn/wiki/en/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Al-Haytham.html">"Abu Ali al-Hasan ibn al-Haytham"</a>, <i><a href="https://www.search.com.vn/wiki/en/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="https://www.search.com.vn/wiki/en/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abu+Ali+al-Hasan+ibn+al-Haytham&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FAl-Haytham.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFMacKayOldford2000" class="citation cs2">MacKay, R. J.; Oldford, R. W. (August 2000), "Scientific Method, Statistical Method and the Speed of Light", <i>Statistical Science</i>, <b>15</b> (3): 254–78, <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Fss%2F1009212817">10.1214/ss/1009212817</a></span>, <a href="https://www.search.com.vn/wiki/en/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1847825">1847825</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=Scientific+Method%2C+Statistical+Method+and+the+Speed+of+Light&rft.volume=15&rft.issue=3&rft.pages=254-78&rft.date=2000-08&rft_id=info%3Adoi%2F10.1214%2Fss%2F1009212817&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1847825%23id-name%3DMR&rft.aulast=MacKay&rft.aufirst=R.+J.&rft.au=Oldford%2C+R.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-Weiss-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Weiss_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFNeumann1998" class="citation cs2"><a href="https://www.search.com.vn/wiki/en/Peter_M._Neumann" title="Peter M. Neumann">Neumann, Peter M.</a> (1998), "Reflections on Reflection in a Spherical Mirror", <i><a href="https://www.search.com.vn/wiki/en/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>105</b> (6): 523–528, <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2589403">10.2307/2589403</a>, <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2589403">2589403</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Reflections+on+Reflection+in+a+Spherical+Mirror&rft.volume=105&rft.issue=6&rft.pages=523-528&rft.date=1998&rft_id=info%3Adoi%2F10.2307%2F2589403&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589403%23id-name%3DJSTOR&rft.aulast=Neumann&rft.aufirst=Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFShabana1995" class="citation book cs1">Shabana, A. A. (1995-12-08). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G2UyBTji18oC&q=Timoshenko-Rayleigh+theory&pg=PA2"><i>Theory of Vibration: An Introduction</i></a>. Springer Science & Business Media. <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/978-0-387-94524-8" title="Special:BookSources/978-0-387-94524-8"><bdi>978-0-387-94524-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Vibration%3A+An+Introduction&rft.pub=Springer+Science+%26+Business+Media&rft.date=1995-12-08&rft.isbn=978-0-387-94524-8&rft.aulast=Shabana&rft.aufirst=A.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG2UyBTji18oC%26q%3DTimoshenko-Rayleigh%2Btheory%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFAude1949" class="citation cs2">Aude, H. T. R. (1949), "Notes on Quartic Curves", <i><a href="https://www.search.com.vn/wiki/en/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>56</b> (3): 165–170, <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2305030">10.2307/2305030</a>, <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2305030">2305030</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Notes+on+Quartic+Curves&rft.volume=56&rft.issue=3&rft.pages=165-170&rft.date=1949&rft_id=info%3Adoi%2F10.2307%2F2305030&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2305030%23id-name%3DJSTOR&rft.aulast=Aude&rft.aufirst=H.+T.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFRees1922" class="citation journal cs1">Rees, E. L. (1922). "Graphical Discussion of the Roots of a Quartic Equation". <i>The American Mathematical Monthly</i>. <b>29</b> (2): 51–55. <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2972804">10.2307/2972804</a>. <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2972804">2972804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Graphical+Discussion+of+the+Roots+of+a+Quartic+Equation&rft.volume=29&rft.issue=2&rft.pages=51-55&rft.date=1922&rft_id=info%3Adoi%2F10.2307%2F2972804&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2972804%23id-name%3DJSTOR&rft.aulast=Rees&rft.aufirst=E.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFLazard1988" class="citation journal cs1">Lazard, D. (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2888%2980015-4">"Quantifier elimination: Optimal solution for two classical examples"</a>. <i>Journal of Symbolic Computation</i>. <b>5</b> (1–2): 261–266. <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2888%2980015-4">10.1016/S0747-7171(88)80015-4</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=Quantifier+elimination%3A+Optimal+solution+for+two+classical+examples&rft.volume=5&rft.issue=1%E2%80%932&rft.pages=261-266&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2FS0747-7171%2888%2980015-4&rft.aulast=Lazard&rft.aufirst=D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0747-7171%252888%252980015-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFDescartes1954" class="citation cs2"><a href="https://www.search.com.vn/wiki/en/Ren%C3%A9_Descartes" title="René Descartes">Descartes, René</a> (1954) [1637], "Book III: On the construction of solid and supersolid problems", <i><a href="https://www.search.com.vn/wiki/en/La_G%C3%A9om%C3%A9trie" title="La Géométrie">The Geometry of Rene Descartes with a facsimile of the first edition</a></i>, <a href="https://www.search.com.vn/wiki/en/Dover_Publications" title="Dover Publications">Dover</a>, <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/0-486-60068-8" title="Special:BookSources/0-486-60068-8"><bdi>0-486-60068-8</bdi></a>, <a href="https://www.search.com.vn/wiki/en/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:51.0020.07">51.0020.07</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Book+III%3A+On+the+construction+of+solid+and+supersolid+problems&rft.btitle=The+Geometry+of+Rene+Descartes+with+a+facsimile+of+the+first+edition&rft.pub=Dover&rft.date=1954&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A51.0020.07%23id-name%3DJFM&rft.isbn=0-486-60068-8&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-Brookfield-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brookfield_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFBrookfield,_G.2007" class="citation journal cs1">Brookfield, G. (2007). <a rel="nofollow" class="external text" href="https://www.maa.org/sites/default/files/Brookfield2007-103574.pdf">"Factoring quartic polynomials: A lost art"</a> <span class="cs1-format">(PDF)</span>. <i><a href="https://www.search.com.vn/wiki/en/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>80</b> (1): 67–70. <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570X.2007.11953453">10.1080/0025570X.2007.11953453</a>. <a href="https://www.search.com.vn/wiki/en/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53375377">53375377</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Factoring+quartic+polynomials%3A+A+lost+art&rft.volume=80&rft.issue=1&rft.pages=67-70&rft.date=2007&rft_id=info%3Adoi%2F10.1080%2F0025570X.2007.11953453&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53375377%23id-name%3DS2CID&rft.au=Brookfield%2C+G.&rft_id=https%3A%2F%2Fwww.maa.org%2Fsites%2Fdefault%2Ffiles%2FBrookfield2007-103574.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFvan_der_Waerden1991" class="citation cs2"><a href="https://www.search.com.vn/wiki/en/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, Bartel Leendert</a> (1991), "The Galois theory: Equations of the second, third, and fourth degrees", <i><a href="https://www.search.com.vn/wiki/en/Moderne_Algebra" title="Moderne Algebra">Algebra</a></i>, vol. 1 (7th ed.), <a href="https://www.search.com.vn/wiki/en/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>, <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/0-387-97424-5" title="Special:BookSources/0-387-97424-5"><bdi>0-387-97424-5</bdi></a>, <a href="https://www.search.com.vn/wiki/en/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0724.12001">0724.12001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Galois+theory%3A+Equations+of+the+second%2C+third%2C+and+fourth+degrees&rft.btitle=Algebra&rft.edition=7th&rft.pub=Springer-Verlag&rft.date=1991&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0724.12001%23id-name%3DZbl&rft.isbn=0-387-97424-5&rft.aulast=van+der+Waerden&rft.aufirst=Bartel+Leendert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFEuler1984" class="citation cs2"><a href="https://www.search.com.vn/wiki/en/Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a> (1984) [1765], "Of a new method of resolving equations of the fourth degree", <i><a href="https://www.search.com.vn/wiki/en/Elements_of_Algebra" title="Elements of Algebra">Elements of Algebra</a></i>, <a href="https://www.search.com.vn/wiki/en/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>, <a href="https://www.search.com.vn/wiki/en/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://www.search.com.vn/wiki/en/Special:BookSources/978-1-4613-8511-0" title="Special:BookSources/978-1-4613-8511-0"><bdi>978-1-4613-8511-0</bdi></a>, <a href="https://www.search.com.vn/wiki/en/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0557.01014">0557.01014</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Of+a+new+method+of+resolving+equations+of+the+fourth+degree&rft.btitle=Elements+of+Algebra&rft.pub=Springer-Verlag&rft.date=1984&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0557.01014%23id-name%3DZbl&rft.isbn=978-1-4613-8511-0&rft.aulast=Euler&rft.aufirst=Leonhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li><li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFFaucette1996" class="citation cs2">Faucette, William M. (1996), "A Geometric Interpretation of the Solution of the General Quartic Polynomial", <i><a href="https://www.search.com.vn/wiki/en/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a></i>, <b>103</b> (1): 51–57, <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2975214">10.2307/2975214</a>, <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2975214">2975214</a>, <a href="https://www.search.com.vn/wiki/en/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1369151">1369151</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=A+Geometric+Interpretation+of+the+Solution+of+the+General+Quartic+Polynomial&rft.volume=103&rft.issue=1&rft.pages=51-57&rft.date=1996&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1369151%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2975214%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2975214&rft.aulast=Faucette&rft.aufirst=William+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></span></li></ol></div></div><h2><span class="mw-headline" id="Further_reading">Further reading</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=22" title="Edit section's source code: Further reading"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFCarpenter,_W.1966" class="citation journal cs1">Carpenter, W. (1966). "On the solution of the real quartic". <i><a href="https://www.search.com.vn/wiki/en/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>. <b>39</b> (1): 28–30. <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2688990">10.2307/2688990</a>. <a href="https://www.search.com.vn/wiki/en/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2688990">2688990</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=On+the+solution+of+the+real+quartic&rft.volume=39&rft.issue=1&rft.pages=28-30&rft.date=1966&rft_id=info%3Adoi%2F10.2307%2F2688990&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2688990%23id-name%3DJSTOR&rft.au=Carpenter%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></li><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFYacoub,_M.D.Fraidenraich,_G.2012" class="citation journal cs1">Yacoub, M.D.; Fraidenraich, G. (July 2012). "A solution to the quartic equation". <i><a href="https://www.search.com.vn/wiki/en/Mathematical_Gazette" class="mw-redirect" title="Mathematical Gazette">Mathematical Gazette</a></i>. <b>96</b>: 271–275. <a href="https://www.search.com.vn/wiki/en/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fs002555720000454x">10.1017/s002555720000454x</a>. <a href="https://www.search.com.vn/wiki/en/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124512391">124512391</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Gazette&rft.atitle=A+solution+to+the+quartic+equation&rft.volume=96&rft.pages=271-275&rft.date=2012-07&rft_id=info%3Adoi%2F10.1017%2Fs002555720000454x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124512391%23id-name%3DS2CID&rft.au=Yacoub%2C+M.D.&rft.au=Fraidenraich%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuartic+function" class="Z3988"></span></li></ul><h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="https://www.search.com.vn/wiki/?lang=en&title=Quartic_function&action=edit&section=23" title="Edit section's source code: External links"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></h2><ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/QuarticFormula">Quartic formula as four single equations</a> at <a href="https://www.search.com.vn/wiki/en/PlanetMath" title="PlanetMath">PlanetMath</a>.</li><li><a rel="nofollow" class="external text" href="http://members.tripod.com/l_ferrari/quartic_equation.htm">Ferrari's achievement</a></li></ul><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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navbar-mini"><ul><li class="nv-view"><a href="https://www.search.com.vn/wiki/en/Template:Polynomials" title="Template:Polynomials"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="https://www.search.com.vn/wiki/en/Template_talk:Polynomials" title="Template talk:Polynomials"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="https://www.search.com.vn/wiki/en/Special:EditPage/Template:Polynomials" title="Special:EditPage/Template:Polynomials"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polynomials_and_polynomial_functions" style="font-size:114%;margin:0 4em"><a href="https://www.search.com.vn/wiki/en/Polynomial" title="Polynomial">Polynomials</a> and <a href="https://www.search.com.vn/wiki/en/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="https://www.search.com.vn/wiki/en/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a href="https://www.search.com.vn/wiki/en/Zero_polynomial" class="mw-redirect" title="Zero polynomial">Zero polynomial (degree undefined or −1 or −∞)</a></li><li><a href="https://www.search.com.vn/wiki/en/Constant_function" title="Constant function">Constant function (0)</a></li><li><a href="https://www.search.com.vn/wiki/en/Linear_function_(calculus)" title="Linear function (calculus)">Linear function (1)</a><ul><li><a href="https://www.search.com.vn/wiki/en/Linear_equation" title="Linear equation">Linear equation</a></li></ul></li><li><a href="https://www.search.com.vn/wiki/en/Quadratic_function" title="Quadratic function">Quadratic function (2)</a><ul><li><a href="https://www.search.com.vn/wiki/en/Quadratic_equation" title="Quadratic equation">Quadratic equation</a></li></ul></li><li><a href="https://www.search.com.vn/wiki/en/Cubic_function" title="Cubic function">Cubic function (3)</a><ul><li><a href="https://www.search.com.vn/wiki/en/Cubic_equation" title="Cubic equation">Cubic equation</a></li></ul></li><li><a class="mw-selflink selflink">Quartic function (4)</a><ul><li><a href="https://www.search.com.vn/wiki/en/Quartic_equation" title="Quartic equation">Quartic equation</a></li></ul></li><li><a href="https://www.search.com.vn/wiki/en/Quintic_function" title="Quintic function">Quintic function (5)</a></li><li><a href="https://www.search.com.vn/wiki/en/Sextic_equation" title="Sextic equation">Sextic equation (6)</a></li><li><a href="https://www.search.com.vn/wiki/en/Septic_equation" title="Septic equation">Septic equation (7)</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By properties</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a href="https://www.search.com.vn/wiki/en/Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">Univariate</a></li><li><a href="https://www.search.com.vn/wiki/en/Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">Bivariate</a></li><li><a href="https://www.search.com.vn/wiki/en/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">Multivariate</a></li><li><a href="https://www.search.com.vn/wiki/en/Monomial" title="Monomial">Monomial</a></li><li><a href="https://www.search.com.vn/wiki/en/Binomial_(polynomial)" title="Binomial (polynomial)">Binomial</a></li><li><a href="https://www.search.com.vn/wiki/en/Trinomial" title="Trinomial">Trinomial</a></li><li><a href="https://www.search.com.vn/wiki/en/Irreducible_polynomial" title="Irreducible polynomial">Irreducible</a></li><li><a href="https://www.search.com.vn/wiki/en/Square-free_polynomial" title="Square-free polynomial">Square-free</a></li><li><a href="https://www.search.com.vn/wiki/en/Homogeneous_polynomial" title="Homogeneous polynomial">Homogeneous</a></li><li><a href="https://www.search.com.vn/wiki/en/Quasi-homogeneous_polynomial" title="Quasi-homogeneous polynomial">Quasi-homogeneous</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools and algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a href="https://www.search.com.vn/wiki/en/Factorization_of_polynomials" title="Factorization of polynomials">Factorization</a></li><li><a href="https://www.search.com.vn/wiki/en/Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">Greatest common divisor</a></li><li><a href="https://www.search.com.vn/wiki/en/Polynomial_long_division" title="Polynomial long division">Division</a></li><li><a href="https://www.search.com.vn/wiki/en/Horner%27s_method" title="Horner's method">Horner's method of evaluation</a></li><li><a href="https://www.search.com.vn/wiki/en/Polynomial_identity_testing" title="Polynomial identity testing">Polynomial identity testing</a></li><li><a href="https://www.search.com.vn/wiki/en/Polynomial_resultant" class="mw-redirect" title="Polynomial resultant">Resultant</a></li><li><a href="https://www.search.com.vn/wiki/en/Discriminant" title="Discriminant">Discriminant</a></li><li><a href="https://www.search.com.vn/wiki/en/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li></ul></div></td></tr></tbody></table></div></div>' |