Problem 44 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{x+3}-\frac{2}{(x+3)^{2}}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{5x+13}{(x+3)^{2}}\)

Step 1: Identify common denominator

Since the second denominator is the square of the first denominator, we can use \((x+3)^{2}\) as the common denominator.

Step 2: Create equivalent fractions

To make the first fraction have \((x+3)^{2}\) as the denominator, multiply the numerator and denominator of the first fraction by \((x+3)\):\[\frac{5}{x+3} * \frac{x+3}{x+3} = \frac{5(x+3)}{(x+3)^{2}}\]This gives us two fractions with the same denominator:\[\frac{5(x+3)}{(x+3)^{2}}-\frac{2}{(x+3)^{2}}\]

Step 3: Subtract fractions

Now that the two fractions have the same denominator, subtract the numerators:\[\frac{5(x+3)-2}{(x+3)^{2}}\]

Step 4: Simplify the result

Expand the numerator, simplify and collect like terms:\[\frac{5x+15-2}{(x+3)^{2}} = \frac{5x+13}{(x+3)^{2}}\]