History
Solving a quartic equation
Special cases
Consider a quartic equation expressed in the form :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, :Evident roots: 1 and −1 and −''k''
Call our quartic polynomial ''Q''(''x''). Since 1 raised to any power is 1, . Thus if , ''Q''(1) = 0 and so ''x'' = 1 is a root of ''Q''(''x''). It can similarly be shown that if , ''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 , and , then ''x'' = −''k'' is a root of the equation. The full quartic can then be factorized this way: : If , and , ''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 : and thus is a biquadratic equation, which is easy to solve: let , so our equation turns to : which is a simple quadratic equation, whose solutions are easily found using the quadratic formula: : When we've solved it (i.e. found these two ''z'' values), we can extract ''x'' from them : : : : If either of the ''z'' solutions were negative or complex numbers, then some of the ''x'' solutions are complex numbers.Quasi-symmetric equations
: 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'', : The first step should be to eliminate the ''x''3 term. To do this, change variables from ''x'' to ''u'', such that : Then : Expanding the powers of the binomials produces : Collecting the same powers of ''u'' yields : Now rename the coefficients of ''u''. Let : The resulting equation is which is a depressed quartic equation. If 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 . If then one of the roots is and the other roots can be found by dividing by , and solving the resulting Cubic equation#Cardano.27s method, depressed cubic equation, : Using reverse substitution we can find our values for .Ferrari's solution
Otherwise, the depressed quartic can be solved by means of a method discovered by=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 : ::(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: : ::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 and is to note that they are dependent. Equation () is a quadratic equation for ''u''. Its solution is : Simplifying, one gets : This is the solution of the depressed quartic, therefore the solutions of the original quartic equation are ::Remember: The two come from the same place in equation (), and should both have the same sign, while the sign of is independent.=Summary of Ferrari's method
= Given the quartic equation : its solution can be found by means of the following calculations: : : : If then :: Otherwise, continue with : : : (either sign of the square root will do) : (there are 3 complex roots, any one of them will do) : : : ::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 : 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 : where P and Q are given by () has the properties that : and where ''α'' and ''β'' are given by (). This means that () has a real root greater than , and therefore that () has a real root greater than . Using this root the term 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 : but this quartic equation is equivalent to the product of two quadratic equations: and Since : then : Let : : 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 : : : and : Therefore : : Equation () can be solved for ''x'' yielding : : 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 : By equating coefficients, this results in the following set of simultaneous equations: : 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 , which can be obtained by substituting for , then , and: : It's now easy to eliminate both and by doing the following: : If we set , then this equation turns into the cubic equation: : which is solved elsewhere. Once you have , then: : 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 for the square root of 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 : If we now set : 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 : which if we make the simplifying assumption that ''b'' = 0, is equal to : 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 : : then : 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 methodReferences
Notes
External links