The quadratic resolvent of the cubic equation

 $\displaystyle y^{3}+py+q=0,$ (1)

where $p$ and $q$ are known complex numbers, is the auxiliary equation

 $z^{2}+qz-\left(\frac{p}{3}\right)^{3}=0$

determining as its roots (http://planetmath.org/Equation)

 $z_{1}=u^{3},\qquad z_{2}=v^{3}$

the numbers $u$ and $v$ whose sum  $y=u+v$  satisfies the equation (1).  See example of solving a cubic equation (http://planetmath.org/exampleofsolvingacubicequation).

Analogically, a quartic equation has a cubic resolvent (resolvent cubic) equation.

Title quadratic resolvent QuadraticResolvent 2014-11-27 16:05:30 2014-11-27 16:05:30 pahio (2872) pahio (2872) 9 pahio (2872) Definition msc 12D10 quadratic resolvent equation CardanosFormulae TchirnhausTransformations cubic resolvent resolvent equation