Discriminant of a Quadratic

What is the discriminant?

The discriminant of a quadratic is closely linked to the Quadratic Formula. For a general quadratic $$ax^2+bx+c$$, the discriminant, $$D$$, is $D=b^2-4ac$ You can easily see where this comes from in the Quadratic Formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$$D>0$$
Two distinct roots
$$D=0$$
One repeated root
$$D<0$$
No real roots

The case $$D<0$$ can be seen to have no real roots because in the quadratic formula, it would result in square rooting a negative value which cannot give a real answer.

Visual examples for various discriminants

$$D>0$$
$$D=0$$
$$D<0$$