Problem 5 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$\frac{x}{x^{2}-25} \cdot \frac{x-5}{x+5}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{x}{x+5}\)

Step 1: Factorize the quadratic expression

The quadratic expression \(x^{2}-25\) can be factorized into \( (x-5)(x+5) \), since \(x^{2}-25\) is a difference of two squares, and can be written as \( (x)^2 - (5)^2 \).

Step 2: Replace the quadratic expression with its factorized form

Now the given problem can be rewritten by replacing the quadratic expression \(x^{2}-25\) with its factorized form \( (x-5)(x+5) \). So it becomes, \(\frac{x}{(x-5)(x+5)} \cdot \frac{(x-5)}{(x+5)}\).

Step 3: Cancel out common factors

In the expression \(\frac{x}{(x-5)(x+5)} \cdot \frac{(x-5)}{(x+5)}\), we see that \(x-5\) and \(x+5\) are present in the numerator and the denominator. These are common factors, and we hence cancel these out. After cancelling, we obtain the simplified expression as \( \frac{x}{x+5}\).