Discriminant and Nature of Roots

When you use the quadratic formula to solve $a{x}^2+bx+c=0$, you might notice a special part hiding underneath the square root sign.

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

The expression $b^2-4ac$ is called the discriminant. It gets its own special symbol — the Greek letter delta ($\Delta$).

The quadratic formula can be rewritten more neatly using $\Delta$:

$x=\frac{-b\pm \sqrt{\Delta }}{2a}$

The value of $\Delta$ determines the nature of the roots — that is, how many real solutions the quadratic equation has. There are exactly three cases:

Case 1: $\Delta >0$ — Two distinct real roots
If $\Delta >0$, then $\sqrt{\Delta }$ is a real, positive number. The $\pm$ in the formula gives two different answers:
$x_1=\frac{-b+\sqrt{\Delta }}{2a}\text{and}{x}_2=\frac{-b-\sqrt{\Delta }}{2a}$

Case 2: $\Delta =0$ — One repeated real root
If $\Delta =0$, then $\sqrt{\Delta }=0$ and the $\pm$ makes no difference. Both solutions collapse into one:
$x=\frac{-b}{2a}$
This is called a repeated root (or a double root).

Case 3: $\Delta <0$ — No real roots
If $\Delta <0$, then $\sqrt{\Delta }$ is the square root of a negative number, which does not exist in the real numbers. The equation has no real solutions.