Problem 53 - Introductory Algebra for College Students

Question

Factor: \(25 x^{2}-81 .\) (Section 6.4, Example 1)

Step-by-Step Solution

Answer

The factored form of \(25x^{2}-81\) is \((5x - 9)(5x + 9)\).

Step 1: Identify the Difference of Squares

A binomial is a difference of squares if it can be written in the form \(a^{2}-b^{2}\), where \(a\) and \(b\) are real numbers. In the expression \(25x^{2}-81\), we can rewrite it as \((5x)^{2} - (9)^{2}\). This confirms that the exercise is a difference of squares.

Step 2: Factor using the Difference of Squares Formula

The formula for factoring the difference of squares is \(a^{2} - b^{2} = (a - b)(a + b)\). Applying this formula on \((5x)^{2} - (9)^{2}\), we get \((5x - 9)(5x + 9)\). This is the factored form of the given expression.