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


x² + 2x - 8 = 0

Verified step by step guidance

1. Start with the quadratic equation given: \(x^{2} + 2x - 8 = 0\).
2. Factor the quadratic expression on the left side. Look for two numbers that multiply to \(-8\) and add to \(2\).
3. Rewrite the quadratic as a product of two binomials: \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
4. Apply the zero-factor property, which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\). Set each binomial equal to zero: \(x + a = 0\) and \(x + b = 0\).
5. Solve each linear equation for \(x\) to find the solutions to the original quadratic equation.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by factoring them into binomials set equal to zero.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. For example, x² + 2x - 8 factors into (x + 4)(x - 2). This step is essential before applying the zero-factor property to find the solutions.