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. Write the factored form as \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
4. Apply the zero-factor property, which states that if \((x + a)(x + b) = 0\), then either \(x + a = 0\) or \(x + b = 0\).
5. Solve each linear equation separately to find the values of \(x\).

Key Concepts

Zero-Factor Property

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is used to solve quadratic equations by factoring the equation into two binomials and setting each equal to zero to find the solutions.

Factoring Quadratic Equations

Factoring involves rewriting a quadratic equation in the form ax² + bx + c = 0 as a product of two binomials. This process requires finding two numbers that multiply to ac and add to b, enabling the equation to be expressed as (x + m)(x + n) = 0.

Solving Quadratic Equations

Once factored, solving quadratic equations involves applying the zero-factor property by setting each factor equal to zero and solving for x. This yields the roots or solutions of the equation, which can be real or complex numbers depending on the discriminant.