The Discriminant: How Many Roots?

≈ 25 min

The expression under the square root, $\Delta = b^2 - 4ac$ , is called the discriminant. It discriminates between three situations — without solving anything:

$\Delta$	Roots
$\Delta > 0$	two distinct real roots
$\Delta = 0$	one repeated real root
$\Delta < 0$	no real roots

Why: the formula says $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ . A positive $\Delta$ gives $\pm$ two different values; zero gives $\pm 0$ (the same value twice); negative has no real square root at all.

Example. $x^2 + 2x + 5 = 0$ : $\Delta = 4 - 20 = -16 < 0$ → no real roots. The parabola never touches the $x$ -axis.

If $\Delta = b^2 - 4ac > 0$ , the quadratic equation has…

no real roots

one repeated real root

two distinct real roots

infinitely many roots

Compute the discriminant of $x^2 + 2x + 5 = 0$ .

A quadratic has $\Delta = 0$ . Geometrically, its parabola…

crosses the $x$ -axis twice

touches the $x$ -axis at exactly one point

never meets the $x$ -axis

is a straight line

Compute the discriminant of $2x^2 - 4x + 2 = 0$ .