Solve each equation using the zero-factor property. See Example 1. x^2 + 2x - 8 = 0

Verified step by step guidance

1. Rewrite the quadratic equation in standard form, which is already given as \(x^2 + 2x - 8 = 0\).
2. Factor the quadratic expression on the left-hand side. Look for two numbers that multiply to \(-8\) (the constant term) and add to \(2\) (the coefficient of the linear term).
3. Once you find the numbers, rewrite the quadratic expression as a product of two binomials: \((x + a)(x + b) = 0\).
4. Apply the zero-factor property, which states that if a product of two factors is zero, then at least one of the factors must be zero. Set each factor 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 principle is essential for solving quadratic equations, as it allows us to set each factor equal to zero to find the solutions. For example, if we have an equation in the form (x - a)(x - b) = 0, we can conclude that x = a or x = b.

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting the equation in a product form, typically as (x - p)(x - q) = 0, where p and q are the roots of the equation. This process simplifies solving the equation by transforming it into a form where the Zero-Factor Property can be applied. In the given equation x^2 + 2x - 8 = 0, we need to find two numbers that multiply to -8 and add to 2.

Quadratic Formula

The Quadratic Formula is a method for solving quadratic equations of the form ax^2 + bx + c = 0, given by x = (-b ± √(b² - 4ac)) / (2a). This formula provides a systematic way to find the roots of any quadratic equation, even when factoring is difficult or impossible. It is particularly useful when the equation does not factor neatly into integers.