Problem 29 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}-25}{2 x-2} \div \frac{x^{2}+10 x+25}{x^{2}+4 x-5}$$

Step-by-Step Solution

Answer

\(\frac{x-5}{2(x+5)}\)

Step 1 - Simplify each fraction

Firstly, try simplifying the given fractions. We can simplify \(\frac{x^{2}-25}{2x-2}\) by factoring and simplifying: \(\frac{(x-5)(x+5)}{2(x-1)}\). Similarly, we can simplify \(\frac{x^{2}+10x+25}{x^{2}+4x-5}\) to: \(\frac{(x+5)^2}{(x-1)(x+5)}\).

Step 2 - Change the operation to multiplication

To divide fractions, it's convenient to change the operation to multiplication by using the reciprocal of the second fraction. So we rewrite the problem as: \(\frac{(x-5)(x+5)}{2(x-1)} \times \frac{(x-1)(x+5)}{(x+5)^2}\).

Step 3 - Cancel out common factors

In the expression \(\frac{(x-5)(x+5)}{2(x-1)} \times \frac{(x-1)(x+5)}{(x+5)^2}\), the factors \((x-1)\), \((x+5)\) in the numerator and denominator can be cancelled to simplify the expression. Doing so, we get: \(\frac{x-5}{2} \times \frac{1}{x+5}\).

Step 4 - Simplify the multiplication of fractions

Finally, multiply the fractions as \(\frac{x-5}{2} \times \frac{1}{x+5}\) which simplifies to \(\frac{x-5}{2(x+5)}\).