Discriminant

Definition

For $ax^2 + bx + c = 0$, the discriminant is

$$\Delta = b^2 - 4ac$$

It is the part sitting under the radical in the quadratic formula.

What its sign tells you

$$\begin{cases} \Delta > 0 & \text{two distinct real solutions} \\[2pt] \Delta = 0 & \text{exactly one real solution} \\[2pt] \Delta < 0 & \text{no real solutions} \end{cases}$$

The reason is the square root. A positive number has two square roots, so the $\pm$ produces two answers. Zero has one. A negative number has no real square root at all, so the formula produces nothing real.

A practical use

Compute $\Delta$ before trying to factor. If it is a perfect square, whole-number factors exist and factoring will work. If it is not, go straight to the formula instead of hunting for a factor pair that is not there.