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