Problem 18 - College Algebra

Question

Multiply or divide as indicated. $$\frac{x^{2}-4}{x^{2}-4 x+4} \cdot \frac{2 x-4}{x+2}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expressions is '2' with a restriction for 'x' that it should not be equal to '2'

Step 1: Factor the polynomials

Start with factoring the polynomials in the numerator and denominator of both expressions. \[ \frac{x^{2}-4}{x^{2}-4 x+4} \cdot \frac{2 x-4}{x+2} = \frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{(x+2)} \]

Step 2: Cancel the common terms

Next, cancel out the common terms in the numerator and the denominator of the resulted expressions. This simplification step is based on the principle that \(\frac{A}{A} = 1\) where \(A ≠ 0\). Also note that, when cancelling common factors, we must also specify the conditions related to the undefined operation \(division by zero\), which originate from the original denominator. Hence, from the expression, \(2(x-2), (x+2), (x-2)^2\) in the denominator haven’t to be equal to zero. We have the following conditions \(x ≠ 2, x ≠ -2\), however in the original equation we have only restriction \(x ≠ 2\), because \(x ≠ -2\) is already cancelled out. Thus, we have \[ \frac{(x-2)(x+2)}{(x-2)^2} \cdot \frac{2(x-2)}{(x+2)} = 2 \], provided that \(x ≠ 2 \)

Step 3: Give the final answer

We write the final simplified result of the given rational expressions. \(2\), \(x ≠ 2\) is our solution