Problem 126 - College Algebra Essentials

Question

Explain how to factor the difference of two squares. Provide an example with your explanation.

Step-by-Step Solution

Answer

The difference of squares \(x^2 - 16\) can be factored into \((x - 4)(x + 4)\).

Step 1: Understanding the formula for the difference of two squares

The formula for the difference of two squares is \(a^2 - b^2 = (a - b)(a + b)\). This formula allows you to take a difference of two squares, such as \(x^2 - 9\), and break it down into \((x - 3)(x + 3)\). Understanding this formula is key as it's applied routinely in factoring tasks.

Step 2: Identifying a difference of squares

Suppose you have an example like \(x^2 - 16\). This is a difference of squares because \(x^2\) is the square of \(x\) and \(16\) is the square of \(4\). Therefore, \(a\) in our formula is equivalent to \(x\) and \(b\) is equivalent to \(4\). Identifying a difference of squares in this way is a crucial step.

Step 3: Applying the formula

After identifying \(a\) and \(b\), you can now apply the formula. Plugging \(x\) and \(4\) into the difference of squares formula gives us \((x - 4)(x + 4)\) as the factored form of \(x^2 - 16\). This step is where the previous steps come together to complete the factoring task.