Ferrari's Method/Also presented as

From ProofWiki
Jump to navigation Jump to search

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

Retrieved from "https://proofwiki.org/w/index.php?title=Ferrari%27s_Method/Also_presented_as&oldid=633913"