Problem 156 - College Algebra

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

Step 1: Factorize the quadratic expression

Start by looking for two numbers that multiply to give +8 (the constant term) and sum to give +6 (the coefficient of \textit{x}). The numbers that satisfy these conditions are +2 and +4. So, the factorized form of the expression becomes \(x^{2}+6 x+8 = (x+2)(x+4)\).

Step 2: Apply the Zero-Product Principle

according to the zero-product principle, if the equation \(ab = 0\), then \(a = 0\) or \(b = 0\). Apply this principle to find the values of \textit{x}, which produces the equations \(x + 2 = 0\) and \(x + 4 = 0\).

Step 3: Solve for x

Solving the two equations from step 2 yields the solutions \(x = -2\) and \(x = -4\). These are the roots of the quadratic equation.