Problem 47 - College Algebra

Question

Add or subtract as indicated. $$\frac{x+5}{x-5}+\frac{x-5}{x+5}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{x^2 + 25}{x^2/2 - 25/2}\)

Step 1: Find the Common Denominator

The denominators in the expression are \(x - 5\) and \(x + 5\). The common denominator can be found by multiplying these two, i.e., \(x^2 - 25\).

Step 2: Rewrite the Fractions with the Common Denominator

By the rule of fractions, when getting to the same denominator, we need to multiply and divide each fraction by what's missing in their denominator. Therefore, rewriting, we get: \(\frac{x+5}{x-5} \cdot \frac{x+5}{x+5} + \frac{x-5}{x+5} \cdot \frac{x-5}{x-5} = \frac{(x+5)^2}{x^2 - 25} + \frac{(x-5)^2}{x^2 - 25}\)

Step 3: Simplify the Expression

Opening the brackets, further simplifying the expression gives: \(\frac{x^2 + 10x + 25 + x^2 - 10x + 25}{x^2 - 25} = \frac{2x^2 + 50}{x^2 - 25}\).

Step 4: Simplify the Expression Further

Dividing the denominator and the numerator by 2, we have this simplified to: \(\frac{x^2 + 25}{x^2/2 - 25/2}\).