Solve each equation using the zero-factor property. x² + 2x - 8 = 0

Verified step by step guidance

1. Start with the given quadratic equation: \(x^{2} + 2x - 8 = 0\).
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 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 equation for \(x\) to find the solutions to the original quadratic equation.

Key Concepts

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 essential for solving quadratic equations that are factored into binomials, allowing us to set each factor equal to zero and solve for the variable.

Factoring Quadratic Equations

Factoring involves rewriting a quadratic equation as a product of two binomials. For example, x² + 2x - 8 can be factored into (x + 4)(x - 2). This step is crucial because it prepares the equation for applying the zero-factor property to find the solutions.

Solving Quadratic Equations

Solving quadratic equations means finding the values of the variable that satisfy the equation. After factoring and applying the zero-factor property, you solve the resulting simple linear equations to find the roots or solutions of the original quadratic.