Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. x^2 - 8x + 16 = 0

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac for a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots of the equation. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, the solutions are nonreal complex numbers.

The solutions of a quadratic equation can be classified based on the value of the discriminant. If the discriminant is positive, the solutions are two distinct rational or irrational numbers, depending on whether the square root of the discriminant is a perfect square. A zero discriminant indicates one rational solution, while a negative discriminant signifies two nonreal complex solutions, which cannot be expressed as real numbers.

Rational numbers are those that can be expressed as the quotient of two integers, while irrational numbers cannot be expressed in such a form and have non-repeating, non-terminating decimal expansions. When evaluating the solutions of a quadratic equation, determining whether the roots are rational or irrational depends on the nature of the discriminant; specifically, if the square root of the discriminant is a perfect square, the solutions are rational.