Problem 51 - College Algebra

Question

Add or subtract as indicated. $$\frac{4}{x^{2}+6 x+9}+\frac{4}{x+3}$$

Step-by-Step Solution

Answer

The simplified sum of the given fractions is: \(\frac{4x+16}{(x+3)^2}\)

Step 1: Identify a Common Denominator

First, recognize that the denominator of the first fraction, \(x^{2}+6x+9\), is a perfect square: \((x+3)^2\). The denominator of the second fraction is \(x+3\), which is the square root of the first fraction's denominator. Therefore, the common denominator of these two fractions is \((x+3)^2\).

Step 2: Express the Fractions with the Common Denominator

The first fraction is already expressed in terms of the common denominator. However, the second fraction isn't. We need to multiply both the numerator and the denominator of the second fraction by \(x+3\) in order to express it in terms of the common denominator. Thus, the second fraction becomes: \(\frac{4(x+3)}{(x+3)^2}\).

Step 3: Perform the Addition

Now that both fractions have the same denominator, we can add them together. This means we add the numerators, while keeping the denominator the same: \(\frac{4+4(x+3)}{(x+3)^2}\).

Step 4: Simplify the Numerator

Perform the operation in the numerator to simplify the expression further: \(\frac{4+4x+12}{(x+3)^2} = \frac{4x+16}{(x+3)^2}\).