Problem 29 - College Algebra Essentials

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

The result of the division operation is \(\frac{(x-5)(x+1)}{2(x+5)(x-1)}\)

Step 1: Simplify the Fractions

Simplify the fractions so as to make operations less complex if possible. Since both \(x^{2}-25\) and \(2 x-2\) in the first fraction can be factored, rewrite the fraction as: \[\frac{(x-5)(x+5)}{2(x-1)}\] The second fraction, \(\frac{x^{2}+10x+25}{x^{2}+4x-5}\), can also be factored to: \[\frac{(x+5)^2}{(x+1)(x-5)}\]

Step 2: Divide the Fractions

Division of fractions can be rewritten as multiplication by the reciprocal of the divisor: \[\frac{(x-5)(x+5)}{2(x-1)} * \frac{(x+1)(x-5)}{(x+5)^2}\]

Step 3: Simplify the Result

Simplify the resulting expression by cancelling out common terms in the numerator and the denominator to achieve the simplest form: \[\frac{(x-5)(x+1)}{2(x+5)(x-1)}\]