Ferrari's Method/Also presented as

Ferrari's Method: Also presented as

Ferrari's method can also be presented in the following form:

Let $P$ be the quartic equation:

$x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0$

Then $P$ has solutions:

$x = \dfrac {-p \pm \sqrt {p^2 - 8 q} } 4$

where:

$\ds p$

$=$

$\ds a_3 \pm \sqrt { {a_3}^2 - 4 a_2 + 4 y_1}$

$\ds q$

$=$

$\ds y_1 \mp \sqrt { {y_1}^2 - 4 a_0}$

where $y_1$ is a real solution to the cubic:

$y^3 - a_2 y^2 + \paren {a_1 a_3 - 4 a_0} y - \paren { {a_1}^2 + a_0 {a_3}^2 - 4 a_0 a_2} = 0$

Source of Name

This entry was named for Lodovico Ferrari.

Historical Note

Ferrari's Method was published by Gerolamo Cardano in $1545$, in his Artis Magnae, Sive de Regulis Algebraicis, on which Lodovico Ferrari collaborated.

Sources

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.8$ Algebraic Equations: Solution of Quartic Equations: $3.8.3$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 9$: Solutions of Algebraic Equations: Quartic Equation: $x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$: $9.6$, $9.7$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 5$: Solutions of Algebraic Equations: Quartic Equation: $x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$: $5.6.$, $5.7.$