Definition:Discriminant of Polynomial/Cubic Equation

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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.


Sources