How to find the discriminant of a quadratic equation and describe the nature of the roots?

In the quadratic ax²+bx+c the discriminant is determined by b²-4ac. If this expression is positive the roots are real; if negative, the roots are complex; if the expression is zero the quadratic is a perfect square of the form x-b/2a.

If b²-4ac is a perfect square and positive the roots are rational, otherwise they are irrational.