# quadratic resolvent

The quadratic resolvent of the cubic equation^{}

${y}^{3}+py+q=0,$ | (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)

$${z}_{1}={u}^{3},{z}_{2}={v}^{3}$$ |

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 |
---|---|

Canonical name | QuadraticResolvent |

Date of creation | 2014-11-27 16:05:30 |

Last modified on | 2014-11-27 16:05:30 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 12D10 |

Synonym | quadratic resolvent equation |

Related topic | CardanosFormulae |

Related topic | TchirnhausTransformations |

Defines | cubic resolvent |

Defines | resolvent equation |