Problem 26 - College Algebra

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 solution to the operation \(\frac{x^{2}-4}{x-2} \div \frac{x+2}{4x-8}\) is \(\frac{1}{4}x + \frac{1}{2}\).

Step 1: Simplify the Fractions

Initially, it's important to break down the algebraic expressions. The polynomial \(x^{2}-4\) can be expressed as \((x-2)(x+2)\) since it's a difference of squares. The operation \(\frac{x^{2}-4}{x-2}\) then simplifies to \(x+2\) after canceling out \(x-2\) in the numerator and the denominator. Similarly, the denominator in the second fraction, \(4x-8\), can be factored as \(4(x-2)\), then \(\frac{x+2}{4x-8}\) simplifies to \(\frac{1}{4}\) after canceling out \(x+2\) and \(x-2\).

Step 2: Convert the Division to Multiplication

The operation \(\frac{x^{2}-4}{x-2} \div \frac{x+2}{4x-8}\) is equivalent to \((x+2) \times \frac{1}{4}\) after converting the division to multiplication and using the reciprocal of the second term.

Step 3: Multiply

After the conversion, the multiplication operation becomes simple. The term \((x+2) \times \frac{1}{4}\) simplifies further to \(\frac{1}{4}x + \frac{1}{2}\).