Problem 156 - Algebra and Trigonometry

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

Step 1: Understanding the Factorization

The first step is to factorize the quadratic equation. Factoring is writing the expression as a product of its factors. Here, the task is to find two numbers when multiplied give 8 (c term) and when added give 6 (b term).

Step 2: Carrying out the Factorization

Since the numbers 4 and 2 fulfill the conditions (4 * 2 = 8 and 4 + 2 = 6), the expression can be factored as \((x + 4)(x + 2) = 0\)

Step 3: Applying the Zero-Product Principle

When a product of factors equals zero, at least one of the factors must be zero. The zero - product principle states that if \((x + 4)(x + 2) = 0\), then either \((x + 4) = 0\) or \((x + 2) = 0\)

Step 4: Solving for 'x'

Solve each of these two equations for 'x'. For \((x + 4) = 0\), subtract 4 from both sides to get \(x= -4\). For \((x + 2) = 0\), subtract 2 from both sides to get \(x= -2\)