The quadratic formula and the discriminant

Let us now look at the general quadratic formula. The discriminant $\Delta$ tells us at once how many real solutions the equation has.

In brief — Quadratic formula

$\boxed{x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}}$

The discriminant $\Delta = b^2-4ac$ determines the nature of the solutions:

• $\Delta > 0$: two distinct real solutions
• $\Delta = 0$: one repeated real solution
• $\Delta < 0$: no real solution

The sign of the discriminant is therefore a quick indicator: it is enough to compute $b^2-4ac$ to know, even before taking the square root, whether and how many real solutions we shall find.

Links

Topics: Quadratic equations
Concepts: Discriminant · Quadratic equation · Quadratic formula
Methods: Delta of the quadratic
Skills: Solving equations · Using formulae