Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 - x - 1 = 0

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the quadratic equation. This formula provides the roots of the equation, which can be real or complex depending on the value of the discriminant (b² - 4ac).

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Understanding the discriminant helps in predicting the type of solutions without solving the equation.