Using Discriminants to Find the Number of Roots of a Quadratic Curve

In general a quadratic curve may have one, two or no roots, as shown.

For the curve the number of roots depends entirely on the discriminant

Ifthere are no roots.

Ifthere is one root.

Ifthere are two roots.

We are typically asked: Find the set of values offor which the curve has

no roots:

always, for every value ofso for every value of k the curve has no roots.

Example: Find the set of values offor which the curvehas two roots.

We solve for

Soor

Example: Find the set of values offor which the curvehas one root.

We solve for

Soor