Definition:Cubic Equation

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Definition

A cubic equation is a polynomial equation of the form:

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


Discriminant

Let:

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

The discriminant of the cubic equation is given by:

$D := Q^3 + R^2$


Resolvent Equation

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 see

  • Results about cubic equations can be found here.


Sources