In mathematics, a quartic equation is a polynomial equation of degree 4.The quartic equations were solved as soon as the methods of solving third degree equations were known. The Ferrari method and the Descartes method were successively developed.
The Ferrari method imagined and perfected by Ludovico Ferrari (1540) makes it possible to solve the equations of the fourth degree by radicals, that is to say to write the solutions as a combination of additions, subtractions, multiplications, divisions , and square, cubic and quartic roots formed from the coefficients of the equation. It provides for the four solutions, under a different appearance, the same formula as that of the later methods of Descartes (1637) and Lagrange (1770).
We first reduce the equation (by dividing by the dominant coefficient then by translating the variable so as to eliminate the term of degree 3) to an equation of the form z4+ pz2 + qz + r = 0. The central point of the method consists in replacing the monomial z4 by the polynomial (z²+λ)²-2λz²-λ², parameterized by λ, and in finding a suitable value of λ, which allows to write z4 + pz² + qz + r as a difference of two squares therefore, via a remarkable identity, as a product of two polynomials of the second degree.
Some authors prefer to start with a completion of the square, z4+ pz²=(z²+p/2)²-p²/4, which allows them to present the Ferrari method with another parameter (u=λ-p/2) , equal to half of that of Descartes and Lagrange (y=2λ-p).