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A method for reducing the solution of an equation of degree 4 to the
solution of one cubic and two quadratic equations; it was discovered
by L. Ferrari (published in 1545).

The Ferrari method for the equation 
$$y^4 + a y^3 + by^2 + cy + d = 0$$
consists in the
following. By the substitution $y=x-a/4$ the given equation can be reduced
to 
$$x^4+px^2 + qx +r = 0,\label{1}$$
which contains no term in $x^3$. If one introduces an auxiliary
parameter $\def\a{\alpha}\a$, the left-hand side of (1) can be written as 
$$x^4+px^2+qx+r =$$

$$=(x^2+\frac{p}{2}+\a)^2-\big[2\a x^2 - qx +\big(\a^2+p\a+\frac{p^2}{4}-r\big)\big].\label{2}$$
One then chooses a value of $\a$ such that the expression in
square brackets is a perfect square. For this the discriminant of the
quadratic trinomial must vanish. This gives a
[[Cubic equation|cubic equation]] for $\a$, 
$$q^2-4\cdot2\a\big(\a^2+p\a+\frac{p^2}{4}-r\big)=0.$$
Let $\a_0$ be one of the
roots of this equation. For $\a=\a_0$ the polynomial in square brackets in
(2) has one double root, 
$$x_0 = \frac{q}{4\a_0},$$
which leads to the equation 
$$(x^2+\frac{p}{2}+\a_0)^2 - 2\a_0(x-x_0)^2 = 0.$$
This
equation of degree 4 splits into two quadratic equations. The roots of
these equations are also the roots of (1).

====References====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
</TR></table>