View source for Ferrari method

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384601.png" /></td> </tr></table>

consists in the following. By the substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384602.png" /> the given equation can be reduced to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

which contains no term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384604.png" />. If one introduces an auxiliary parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384605.png" />, the left-hand side of (1) can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384607.png" /></td> </tr></table>

One then chooses a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384608.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f0384609.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f03846010.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f03846011.png" /> be one of the roots of this equation. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f03846012.png" /> the polynomial in square brackets in (2) has one double root,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f03846013.png" /></td> </tr></table>

which leads to the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038460/f03846014.png" /></td> </tr></table>

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>