Problem 39 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2 x}{x^{2}-16}+\frac{x}{x-4}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^2+6x}{(x+4)(x-4)} \).

Step 1: Factorize the Denominator of the First Fraction

Recognize that the denominator of the first fraction, \(x^{2}-16\), is a difference of two squares. It can be factorized into \((x+4)(x-4)\). Then, rewrite the first fraction as \(\frac{2x}{(x+4)(x-4)}\).

Step 2: Find the Common Denominator

Now take a look at the two fractions. The first fraction has a denominator of \((x+4)(x-4)\) and the second fraction has a denominator of \(x-4\). So, to simplify their addition, they need to have the same denominator. Here, the least common denominator (LCD) is \((x+4)(x-4)\). Multiply the second fraction by \(\frac{x+4}{x+4}\) to achieve the common denominator.

Step 3: Add the Two Fractions

Now that both fractions have the same denominator, they can be added together: \[\frac{2x}{(x+4)(x-4)}+ \frac{x(x+4)}{(x+4)(x-4)}\] Simplify the fractions by adding the numerators and keeping the same denominator. Your result is \(\frac{2x+x(x+4)}{(x-4)(x+4)}\)

Step 4: Simplify the numerator

Expand the expression in the numerator and simplify. This results in \(\frac{2x+x^2+4x}{(x-4)(x+4)}\), which also simplifies to become \(\frac{x^2+6x}{(x-4)(x+4)}\).