Solve each equation using the quadratic formula. See Examples 5 and 6. -4x^2 = -12x + 11

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. Understanding the structure of quadratic equations is essential for solving them effectively.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the roots of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a). This formula provides a systematic way to calculate the solutions, regardless of whether the quadratic can be factored easily, making it a powerful tool in algebra.

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; and if it is negative, there are two complex roots. Understanding the discriminant helps predict the type of solutions before calculating them.