Discriminant of Quadratic Polynomial

Theorem

As a general case of the discriminant of a polynomial, the discriminant of the polynomial \(aX^2 + bX + c\) is given by

\[ b^2 - 4ac.\]

This calculation forms the basis for the characterisation of roots of quadratic by discriminant.

Proof

Consider the polynomial \(aX^2 + bX + c\) with roots \(\alpha_1\) and \(\alpha_2\). We compute the discriminant from the definition and then apply Vieta's formulas:

\[\begin{align*} \mathrm{disc}(aX^2 + bX + c) &= a^{2 \times 2 - 2} (\alpha_1 - \alpha_2)^2 \\ &= a^2(\alpha_1^2 + \alpha_2^2 - 2\alpha_1\alpha_2) \\ &= a^2((\alpha_1 + \alpha_2)^2 - 4\alpha_1\alpha_2) \\ &= a^2\left(\left(-\frac{b}{a}\right)^2 - 4\frac{c}{a}\right) \\ &= a^2\frac{b^2- 4ac}{a^2} \\ &= b^2- 4ac \\ \end{align*}\]