Definition

The discriminant is the expression under the square root in the quadratic formula, $D = b^2 - 4ac$, and it tells you how many real solutions a quadratic $ax^2 + bx + c = 0$ has without fully solving it: $D > 0$ gives two distinct real roots, $D = 0$ gives one repeated (double) root, and $D < 0$ gives two complex conjugate roots, with no real solutions. When $D$ is a perfect square and $a$, $b$, $c$ are integers, the roots are rational. More generally, the discriminant of a degree-$n$ polynomial $f$ is $\Delta(f) = a_n^{2n-2} \prod_{i < j} (r_i - r_j)^2$ over its complex roots $r_i$ (which reduces to $b^2 - 4ac$ for a quadratic and to $-4p^3 - 27q^2$ for a cubic $x^3 + px + q$); it is zero exactly when $f$ has a repeated root, equivalently when $\gcd(f, f')$ is not constant.