Solve each equation using the quadratic formula. See Examples 5 and 6. (4x - 1)(x + 2) = 4x

Key Concepts

Quadratic Formula

The quadratic formula is a method for solving quadratic equations of the form ax² + bx + c = 0. It is expressed as x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are coefficients from the equation. This formula provides the solutions (roots) of the equation, which can be real or complex numbers depending on the value of the discriminant (b² - 4ac).

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions. In the context of the given equation, (4x - 1)(x + 2) = 4x can be expanded and rearranged to form a standard quadratic equation. Understanding how to factor expressions is crucial for simplifying equations and finding solutions efficiently.

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, the roots are complex. Analyzing the discriminant helps in understanding the solutions' characteristics.