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:

\(\ds Q\)

\(=\)

\(\ds \dfrac {3 a c - b^2} {9 a^2}\)

\(\ds R\)

\(=\)

\(\ds \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}\)

The discriminant of the cubic equation is given by:

$D := Q^3 + R^2$

Reduced Form

Let $P$ be a cubic equation expressed in the form:

$x^3 + p x^2 + q x + r = 0$

The discriminant of $P$ is given by:

$D = 4 q^3 + 4 p^3 r + 27 r^3 - p^2 q^2 - 18 p q r$

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

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 9$: Solutions of Algebraic Equations: Cubic Equation: $x^3 + a_1 x^2 + a_2 x + a_3 = 0$: $9.3$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 5$: Solutions of Algebraic Equations: Cubic Equation: $x^3 + a_1 x^2 + a_2 x + a_3 = 0$: $5.3.$