The Discriminant and Complex Roots
The expression , which appears under the radical sign in the quadratic formula
is called the discriminant of the quadratic equation .
If , and are real numbers, you can use the algebraic sign of the discriminant to determine the number and the nature of the roots of the quadratic equation.

If is positive, the equation has two real and unequal roots.

If , the equation has only one root, i.e., double roots.

If is negative, the equation has no real root. Its roots are two complex numbers that are complex conjugates of each other.
Example:
Use the discriminant to determine the nature of the roots of each quadratic equation without actually solving it.
(a)
(b)
(c)
Solution:
(a) Here , , and is positive, hence there are two unequal real roots.
(b) Here , , and , hence the equation has just one root: a double root and this root is a real number.
(c) Here , , and is negative, hence the equation has no real root.
Example:
Use the quadratic formula to find the roots of the quadratic equation .
Solution:
Using the quadratic formula, with , and , we have
Thus, the roots are the complex conjugates and