## You are here

Homequadratic resolvent

## Primary tabs

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

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

the numbers $u$ and $v$ whose sum $y=u+v$ satisfies the equation (1). See example of solving a cubic equation.

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

Defines:

cubic resolvent, resolvent equation

Keywords:

resolvent

Related:

CardanosFormulae, TchirnhausTransformations

Synonym:

quadratic resolvent equation

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

12D10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections