Problem 26 - College Algebra Essentials

Question

Multiply or divide as indicated. $$\frac{x^{2}-4}{x-2} \div \frac{x+2}{4 x-8}$$

Step-by-Step Solution

Answer

The answer is \(4x + 8\).

Step 1: Factorization

The first step is to factorize expression \(x^{2} - 4 = (x - 2)(x + 2)\), and \(4x - 8 = 4(x - 2)\). The expression becomes \(\frac{(x - 2)(x + 2)}{x - 2} \div \frac{x+2}{4(x - 2)}\).

Step 2: Cancel Like Terms

Now we cancel out like terms. The \(x - 2\) on the numerator and denominator cancel out in the first fraction as well as the \(x + 2\) on the numerator and denominator in the second fraction. The expression becomes \(\frac{x + 2}{1} \div \frac{1}{4}\).

Step 3: Convert the Division to Multiplication

Turn the division into a multiplication problem by flipping the second number (reciprocal). The expression becomes \((x + 2) \times 4\).

Step 4: Final Calculation

Now it's just a simple multiplication: 4 times \(x + 2\) equals to \(4x + 8\).