In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. x^2 - 4x - 5 = 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 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, there are two complex solutions.

Quadratic Equation

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 solutions to a quadratic equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula, which incorporates the discriminant.

Types of Solutions

The types of solutions for a quadratic equation are classified based on the value of the discriminant. Real solutions occur when the discriminant is non-negative, leading to either two distinct solutions or one repeated solution. Complex solutions arise when the discriminant is negative, indicating that the roots are not real numbers but rather involve imaginary components.