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. 8x^2 - 72 = 0

Key Concepts

Discriminant

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 provides critical information about the nature of the roots of the equation. Specifically, 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.

Types of Solutions

The solutions of a quadratic equation can be classified based on the value of the discriminant. Real solutions can be either rational or irrational, depending on whether the square root of the discriminant is a perfect square. Nonreal complex solutions occur when the discriminant is negative, indicating that the roots involve imaginary numbers, which cannot be represented on the real number line.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola, and its shape and position are influenced by the coefficients a, b, and c. Understanding the structure of quadratic equations is essential for evaluating their discriminants and determining the nature of their solutions.