Problem 111 - Introductory Algebra for College Students

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 is a simple algebraic expression of the form \(a^2 - b^2\) and can be easily factored to \((a-b)(a+b)\). For example factors of \(x^2 - 9\) are \((x - 3)(x + 3)\).

Step 1: Understand the formula

The difference of squares is an algebraic expression taking the form \(a^2 - b^2\). It's important to recognise this special form because it can be factored easily. The factored form is \((a-b)(a+b)\). It's so because \((a-b)(a+b) = a^2 - b^2\).

Step 2: Consider an example

Now consider an example like \(x^2 - 9\). This is a difference of squares because it is expressed as the difference of the square of \(x\) and the square of 3.

Step 3: Apply the formula

Take the two components from the original equation: \(a = x\) and \(b = 3\). Apply the formula \((a-b)(a+b)\) to get \((x - 3)(x + 3)\). Therefore \(x^2 - 9\) can be factored as \((x - 3)(x + 3)\)