The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

Substitute in the values of , , and .

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant :

means there are distinct real roots.

means there are equal real roots, or distinct real root.

means there are no real roots, but complex roots.

Since the discriminant is less than there are no real roots. Instead, there are two complex roots.

Two Complex Roots

Determine the Nature of the Roots Using the Discriminant x^2-5x+7=0