Problem 89 - Introductory Algebra for College Students

Question

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Step-by-Step Solution

Answer

The solutions of the equation \(x^{2}+6 x+8=0\) are \(x=-2\) and \(x=-4\).

Step 1: Factoring

First, express the quadratic equation \(x^{2}+6 x+8\) in factored form. The factors of \(8\) that add up to \(6\) are \(2\) and \(4\). So, we can express the equation as \((x+2)(x+4)=0\)

Step 2: Applying the zero-product principle

The zero-product property states that if the product of two numbers is zero, then at least one of the numbers must be zero. Based on this principle, we can set each factor to zero to solve for \(x\): \(x+2=0\) and \(x+4=0\)

Step 3: Solving for x

We solve each equation we found in Step 2 separately to find the values of \(x\). The solutions are: \(x=-2\) and \(x=-4\). These are the solutions of the initial quadratic equation.