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mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a quartic equation is one which can be expressed as a ''
quartic function In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A ''qua ...

quartic function
'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one where the coefficients can take any value).


History

Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Ital ...
is attributed 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 Cubic may refer to: Science and mathematics * Cube (algebra) In arithmetic and algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathema ...

cubic
to be found, it couldn't be published immediatel

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'' (1545). The proof that this was the highest order general polynomial for which such solutions could 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.


Solving a quartic equation


Special cases

Consider a quartic equation expressed in the form a_0x^4+a_1x^3+a_2x^2+a_3x+a_4=0 :


Degenerate case

If the constant term ''a''4 = 0, then one of the roots is ''x'' = 0, and the other roots can be found by dividing by ''x'', and solving the resulting cubic equation, :a_0x^3+a_1x^2+a_2x+a_3=0. \,


Evident roots: 1 and −1 and −''k''

Call our quartic polynomial ''Q''(''x''). Since 1 raised to any power is 1, Q(1)=a_0+a_1+a_2+a_3+a_4. Thus if a_0+a_1+a_2+a_3+a_4=0, ''Q''(1) = 0 and so ''x'' = 1 is a root of ''Q''(''x''). It can similarly be shown that if a_0+a_2+a_4=a_1+a_3, ''x'' = −1 is a root. In either case the full quartic can then be divided by the factor (''x'' − 1) or (''x'' + 1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots. If a_1 = a_0 k , a_2 = 0 and a_4= a_3 k, then ''x'' = −''k'' is a root of the equation. The full quartic can then be factorized this way: :a_0 x^4+ a_0 k x^3 + a_3 x + a_3 k = a_0 x^3 (x +k ) + a_3 (x+k) = (a_0 x^3 + a_3) (x+k). \, If a_1 = a_0 k , a_3 = a_2 k and a_4 = 0, ''x'' = 0 and ''x'' = −''k'' are two known roots. ''Q''(''x'') divided by ''x''(''x'' + ''k'') is a quadratic polynomial.


Biquadratic equations

A quartic equation where ''a''3 and ''a''1 are equal to 0 takes the form :a_0x^4+a_2x^2+a_4=0\,\! and thus is a biquadratic equation, which is easy to solve: let z=x^2, so our equation turns to :a_0z^2+a_2z+a_4=0\,\! which is a simple quadratic equation, whose solutions are easily found using the quadratic formula: :z=\frac \,\! When we've solved it (i.e. found these two ''z'' values), we can extract ''x'' from them :x_1=+\sqrt\,\! :x_2=-\sqrt\,\! :x_3=+\sqrt\,\! :x_4=-\sqrt\,\! If either of the ''z'' solutions were negative or complex numbers, then some of the ''x'' solutions are complex numbers.


Quasi-symmetric equations

: a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2=0 \, Steps: # Divide by ''x'' 2. # Use variable change ''z'' = ''x'' + ''m''/''x''.


The general case, along Ferrari's lines

To begin, the quartic must first be converted to a ''depressed quartic''.


Converting to a depressed quartic

Let be the general quartic equation which it is desired to solve. Divide both sides by ''A'', : x^4 + x^3 + x^2 + x + = 0. The first step should be to eliminate the ''x''3 term. To do this, change variables from ''x'' to ''u'', such that : x = u - . Then : \left( u - \right)^4 + \left( u - \right)^3 + \left( u - \right)^2 + \left( u - \right) + = 0. Expanding the powers of the binomials produces : \left( u^4 - u^3 + - + \right) + \left( u^3 - + - \right) + \left( u^2 - + \right) + \left( u - \right) + = 0. Collecting the same powers of ''u'' yields : u^4 + \left( + \right) u^2 + \left( - + \right) u + \left( + - + \right) = 0. Now rename the coefficients of ''u''. Let :\begin \alpha & = + ,\\ \beta & = - + ,\\ \gamma & = + - + . \end The resulting equation is which is a depressed quartic equation. If \beta=0 \ then we have a Quartic equation#Biquadratic equations, biquadratic equation, which (as explained above) is easily solved; using reverse substitution we can find our values for x. If \gamma=0 \ then one of the roots is u=0 and the other roots can be found by dividing by u, and solving the resulting Cubic equation#Cardano.27s method, depressed cubic equation, : u^3 + \alpha u + \beta = 0 \,. Using reverse substitution we can find our values for x.


Ferrari's solution

Otherwise, the depressed quartic can be solved by means of a method discovered by
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Ital ...
. Once the depressed quartic has been obtained, the next step is to add the valid identity : \left(u^2 + \alpha\right)^2 - u^4 - 2 \alpha u^2 = \alpha^2 to equation (), yielding The effect has been to fold up the ''u''4 term into a square number, perfect square: (''u''2 + α)2. The second term, ''αu''2 did not disappear, but its sign has changed and it has been moved to the right side. The next step is to insert a variable ''y'' into the perfect square on the left side of equation (), and a corresponding 2''y'' into the coefficient of ''u''2 in the right side. To accomplish these insertions, the following valid formulas will be added to equation (), : \begin (u^2+\alpha+y)^2-(u^2+\alpha)^2 & = 2y(u^2+\alpha)+ y^2\ \ \\ & = 2yu^2+2y\alpha+y^2, \end and : 0 = (\alpha + 2 y) u^2 - 2 y u^2 - \alpha u^2\, These two formulas, added together, produce : \left(u^2 + \alpha + y\right)^2 - \left(u^2 + \alpha\right)^2 = \left(\alpha + 2 y\right) u^2 - \alpha u^2 + 2 y \alpha + y^2 \qquad \qquad (y\hbox)\, which added to equation () produces : \left(u^2 + \alpha + y\right)^2 + \beta u + \gamma = \left(\alpha + 2 y\right) u^2 + \left(2 y \alpha + y^2 + \alpha^2\right).\, This is equivalent to The objective now is to choose a value for ''y'' such that the right side of equation () becomes a perfect square. This can be done by letting the discriminant of the quadratic function become zero. To explain this, first expand a perfect square so that it equals a quadratic function: : \left(s u + t\right)^2 = \left(s^2\right) u^2 + \left(2 s t\right) u + \left(t^2\right).\, The quadratic function on the right side has three coefficients. It can be verified that squaring the second coefficient and then subtracting four times the product of the first and third coefficients yields zero: : \left(2 s t\right)^2 - 4 \left(s^2\right) \left(t^2\right) = 0.\, Therefore to make the right side of equation () into a perfect square, the following equation must be solved: : (-\beta)^2 - 4 \left(2 y + \alpha\right) \left(y^2 + 2 y \alpha + \alpha^2 - \gamma\right) = 0.\, Multiply the binomial with the polynomial, : \beta^2 - 4 \left(2 y^3 + 5 \alpha y^2 + \left(4 \alpha^2 - 2 \gamma\right) y + \left(\alpha^3 - \alpha \gamma\right)\right) = 0\, Divide both sides by −4, and move the −''β''2/4 to the right, : 2 y^3 + 5 \alpha y^2 + \left( 4 \alpha^2 - 2 \gamma \right) y + \left( \alpha^3 - \alpha \gamma - \frac \right) = 0 Divide both sides by 2, This is a cubic equation in ''y''. Solve for ''y'' using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.


=Folding the second perfect square

= With the value for ''y'' so selected, it is now known that the right side of equation () is a perfect square of the form :\left(s^2\right)u^2 + (2st)u + \left(t^2\right) = \left(\left(\sqrt\right)u + \right)^2 ::(This is correct for both signs of square root, as long as the same sign is taken for both square roots. A ± is redundant, as it would be absorbed by another ± a few equations further down this page.) so that it can be folded: : (\alpha + 2 y) u^2 + (- \beta) u + \left(y^2 + 2 y \alpha + \alpha^2 - \gamma \right) = \left( \left(\sqrt\right)u + \right)^2. ::Note: If ''β'' ≠ 0 then ''α'' + 2''y'' ≠ 0. If ''β'' = 0 then this would be a biquadratic equation, which we solved earlier. Therefore equation () becomes Equation () has a pair of folded perfect squares, one on each side of the equation. The two perfect squares balance each other. If two squares are equal, then the sides of the two squares are also equal, as shown by: Collecting like powers of ''u'' produces ::Note: The subscript ''s'' of \pm_s and \mp_s is to note that they are dependent. Equation () is a quadratic equation for ''u''. Its solution is :u=\frac. Simplifying, one gets :u=. This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are ::Remember: The two \pm_s come from the same place in equation (), and should both have the same sign, while the sign of \pm_t is independent.


=Summary of Ferrari's method

= Given the quartic equation : A x^4 + B x^3 + C x^2 + D x + E = 0, \, its solution can be found by means of the following calculations: : \alpha = - + , : \beta = - + , : \gamma = - + - + . If \,\beta=0, then ::x=-\pm_s\sqrt\qquad \mbox \beta=0 \mbox. Otherwise, continue with : P = - - \gamma, : Q = - + - , : R = - \pm \sqrt, (either sign of the square root will do) : U = \sqrt[3], (there are 3 complex roots, any one of them will do) : y = - \alpha + \beginU=0 &\to -\sqrt[3]\\U\ne 0, &\to U - ,\end \quad\quad\quad :W=\sqrt : x = - + . ::The two ±s must have the same sign, the ±''t'' is independent. To get all roots, compute ''x'' for ±''s''''t'' = +,+ and for +,−; and for −,+ and for −,−. This formula handles repeated roots without problem. Ferrari was the first to discover one of these labyrinthine solutions. The equation which he solved was : x^4 + 6 x^2 - 60 x + 36 = 0 which was already in depressed form. It has a pair of solutions which can be found with the set of formulas shown above.


=Ferrari's solution in the special case of real coefficients

= If the coefficients of the quartic equation are real then the nested depressed cubic equation () also has real coefficients, thus it has at least one real root. Furthermore the cubic function : C(v) = v^3 + P v + Q, where P and Q are given by () has the properties that : C\left(\right) = < 0 and \lim_ C(v) = \infty, where ''α'' and ''β'' are given by (). This means that () has a real root greater than \alpha \over 3, and therefore that () has a real root greater than -\alpha \over 2. Using this root the term \sqrt in () is always real, which ensures that the two quadratic equations () have real coefficients.Carstensen, Jens, ''Komplekse tal, First Edition'', (Systime 1981), .


Obtaining alternative solutions the hard way

It could happen that one only obtained one solution through the formulae above, because not all four sign patterns are tried for four solutions, and the solution obtained is complex number, complex. It may also be the case that one is only looking for a real solution. Let ''x''1 denote the complex solution. If all the original coefficients ''A'', ''B'', ''C'', ''D'' and ''E'' are real—which should be the case when one desires only real solutions – then there is another complex solution ''x''2 which is the complex conjugate of ''x''1. If the other two roots are denoted as ''x''3 and ''x''4 then the quartic equation can be expressed as : (x - x_1) (x - x_2) (x - x_3) (x - x_4) = 0, \, but this quartic equation is equivalent to the product of two quadratic equations: and Since : x_2 = x_1^\star then : \begin (x-x_1)(x-x_2)&=x^2-(x_1+x_1^\star)x+x_1x_1^\star \\ &=x^2-2\operatorname(x_1)x+[\operatorname(x_1)]^2+[\operatorname(x_1)]^2. \end Let : a = - 2\operatorname(x_1), : b = \left[ \operatorname( x_1) \right]^ + \left[ \operatorname(x_1) \right]^ so that equation () becomes Also let there be (unknown) variables ''w'' and ''v'' such that equation () becomes Multiplying equations () and () produces Comparing equation () to the original quartic equation, it can be seen that : a + w = , : b + w a + v = , : w b + v a = , and : v b = . Therefore : w = - a = + 2 \operatorname(x_1), : v = = \frac. Equation () can be solved for ''x'' yielding : x_3 = , : x_4 = . One of these two solutions should be the desired real solution.


Alternative methods


Quick and memorable solution from first principles

Most textbook solutions of the quartic equation require a magic substitution that is almost impossible to memorize. Here is a way to approach it that makes it easy to understand. The job is done if we can factor the quartic equation into a product of two quadratic equation, quadratics. Let : \begin 0 &= x^4 + bx^3 + cx^2 + dx + e \\ &= \left(x^2 + px + q\right)\left(x^2 + rx + s\right) \\ &= x^4 + (p + r)x^3 + (q + s + pr)x^2 + (ps + qr)x + qs \end By equating coefficients, this results in the following set of simultaneous equations: : \begin b & = p + r \\ c & = q + s + pr \\ d & = ps + qr \\ e & = qs \end This is harder to solve than it looks, but if we start again with a Quartic equation#Converting to a depressed quartic, depressed quartic where b = 0, which can be obtained by substituting (x - b/4) for x, then r = -p, and: : \begin c + p^2 & = s + q \\ d/p & = s - q \\ e & = sq \end It's now easy to eliminate both s and q by doing the following: : \begin \left(c + p^2\right)^2 - (d/p)^2 & = (s + q)^2 - (s - q)^2 \\ & = 4sq \\ & = 4e \end If we set P = p^2, then this equation turns into the cubic equation: :P^3 + 2cP^2 + \left(c^2 - 4e\right)P - d^2 = 0 which is solved elsewhere. Once you have p, then: : \begin r & = -p \\ 2s & = c + p^2 + d/p \\ 2q & = c + p^2 - d/p \end The symmetries in this solution are easy to see. 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 p for the square root of P merely exchanges the two quadratics with one another.


Galois theory and factorization

The symmetric group ''S''4 on four elements has the Klein four-group as a normal subgroup. This suggests using a resolvent whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots. Suppose ''r''''i'' for ''i'' from 0 to 3 are roots of :x^4 + bx^3 + cx^2 + dx + e = 0\qquad (1) If we now set : \begin s_0 &= \tfrac12(r_0 + r_1 + r_2 + r_3), \\ s_1 &= \tfrac12(r_0 - r_1 + r_2 - r_3), \\ s_2 &= \tfrac12(r_0 + r_1 - r_2 - r_3), \\ s_3 &= \tfrac12(r_0 - r_1 - r_2 + r_3), \end then since the transformation is an Involution (mathematics), involution, we may express the roots in terms of the four si in exactly the same way. Since we know the value ''s''0 = −''b''/2, we really only need the values for s1, s2 and s3. These we may find by expanding the polynomial :\left(z^2 - s_1^2\right)\left(z^2-s_2^2\right)\left(z^2-s_3^2\right)\qquad (2) which if we make the simplifying assumption that ''b'' = 0, is equal to :z^6 + 2cz^4 + \left(c^2-4e\right) z^2 - d^2 \qquad(3) This polynomial is of degree six, but only of degree three in z2, and so the corresponding equation is solvable. By trial we can determine which three roots are the correct ones, and hence find the solutions of the quartic. We can remove any requirement for trial by using a root of the same resolvent polynomial for factoring; if w is any root of (3), and if : F_1 = x^2+wx+\frac 1 2 w^2+\frac 1 2 c - \frac 1 2\cdot \frac -\frac 1 2 \cdot\frac - \frac + 2\frac : F_2 = x^2-wx + \frac 1 2 w^2 + \frac 1 2 c + \frac 1 2\cdot \frac + \frac - 2\frac + \frac 1 2\cdot \frac then :F_1 F_2 = x^4 + cx^2 + dx + e\qquad\qquad (4) We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.


See also

*Linear equation *Quadratic equation *Cubic equation *Quintic equation *Polynomial *Newton's method


References


Ferrari's achievement
*


Notes


External links



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