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 ri for i from 0 to 3 are roots of If we now set then since the transformation is an involution, we may express the roots in terms of the four si in exactly the same way. Since we know the value s0 = −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. Approximate methods The methods described above are, in principle, exact methods which find the roots once and for all. It is also possible to use methods which give successive approximations which hopefully improve with each iteration. Once such method is the Durand–Kerner method. Such methods may be the only ones available, other than special cases, when trying to solve quintic and higher equations. See also Linear equation Quadratic equation Cubic equation Quintic equation Polynomial Newton's method References Ferrari's achievement Notes External links Calculator for solving Quartics Elementary algebra Equations Polynomials