Definition:Discriminant of Polynomial/Cubic Equation

Theorem
Let $P$ be the cubic equation:
 * $a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$

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

Also see
Note that this is a special case of the general discriminant, although it is important to note that the general formula is given for monic polynomials.