Solve each quadratic equation using the zero-factor property. See Example 5.


x² + 2x - 8 = 0

Verified step by step guidance

1. First, factorize the quadratic equation x² + 2x - 8. Look for two numbers that multiply to -8 (the constant term) and add up to 2 (the coefficient of the linear term x).
2. Rewrite the middle term (2x) using the two numbers found in the previous step, and then group the terms into two pairs.
3. Factor out the greatest common factor from each pair.
4. Set each factor equal to zero, as per the zero-factor property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
5. Solve each resulting simple linear equation to find the values of x.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations 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.

Zero-Product Property

The zero-product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving quadratic equations that have been factored into the form (x - p)(x - q) = 0, allowing us to set each factor equal to zero to find the solutions for x.

Factoring Quadratics

Factoring quadratics involves rewriting the quadratic equation in a product form, typically as (x - p)(x - q) = 0, where p and q are the roots of the equation. This method simplifies the process of finding the solutions by allowing the application of the zero-product property. Mastery of factoring techniques is essential for efficiently solving quadratic equations.