# Definition:Cubic Equation/Resolvent

## Definition

Let $P$ be the cubic equation:

$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$

Let:

$y = x + \dfrac b {3 a}$
$Q = \dfrac {3 a c - b^2} {9 a^2}$
$R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$

Let $y = u + v$ where $u v = -Q$.

The resolvent equation of the cubic is given by:

$u^6 - 2 R u^3 - Q^3$

## Also defined as

Some sources introduce Cardano's Formula starting from the cubic:

$x^3 + q x - r = 0$

to which the general cubic can be reduced to using the Tschirnhaus Transformation.

In this form, the resolvent equation of the cubic is given by:

$u^6 - r u^3 - \dfrac q {27}$