Question

find the sum.
\\(\frac{r}{r^2 - q^2} + \frac{5}{r + q}\\)
\\(\bigcirc\\) a. \\(\frac{r + 5}{r + q}\\)
\\(\bigcirc\\) b. \\(\frac{r + 5}{r^2 - q^2}\\)
\\(\bigcirc\\) c. \\(\frac{6r - q}{r^2 - q^2}\\)
\\(\bigcirc\\) d. \\(\frac{6r - 5q}{r^2 - q^2}\\)

Explanation:

Step1: Factor the denominator

Notice that \( r^2 - q^2 \) is a difference of squares, so \( r^2 - q^2=(r + q)(r - q) \).

Step2: Find a common denominator

The first fraction has denominator \( (r + q)(r - q) \), and the second fraction has denominator \( r + q \). So the common denominator is \( (r + q)(r - q)=r^2 - q^2 \).
Rewrite the second fraction with the common denominator: \( \frac{5}{r + q}=\frac{5(r - q)}{(r + q)(r - q)}=\frac{5r - 5q}{r^2 - q^2} \).

Step3: Add the fractions

Now add the first fraction \( \frac{r}{r^2 - q^2} \) and the rewritten second fraction:
\( \frac{r}{r^2 - q^2}+\frac{5r - 5q}{r^2 - q^2}=\frac{r + 5r - 5q}{r^2 - q^2}=\frac{6r - 5q}{r^2 - q^2} \).

Answer:

D. \(\frac{6r - 5q}{r^2 - q^2}\)