To simplify the given expression [tex]\(\frac{2}{x+7} + \frac{7}{x-7}\)[/tex], follow these steps:
1. Find a common denominator:
The common denominator of [tex]\(x+7\)[/tex] and [tex]\(x-7\)[/tex] is [tex]\((x+7)(x-7)\)[/tex], which can be expanded to [tex]\(x^2 - 49\)[/tex].
2. Rewrite each fraction with the common denominator:
Convert each fraction so that they both have the same denominator [tex]\(x^2 - 49\)[/tex].
[tex]\[
\frac{2}{x+7} \text { becomes } \frac{2(x-7)}{(x+7)(x-7)}
\][/tex]
[tex]\[
\frac{7}{x-7} \text { becomes } \frac{7(x+7)}{(x+7)(x-7)}
\][/tex]
3. Expand the numerators:
Expand the numerators of each fraction:
[tex]\[
\frac{2(x-7)}{(x^2 - 49)} = \frac{2x - 14}{x^2 - 49}
\][/tex]
[tex]\[
\frac{7(x+7)}{(x^2 - 49)} = \frac{7x + 49}{x^2 - 49}
\][/tex]
4. Add the fractions:
Since the denominators are now the same, add the numerators together over the common denominator:
[tex]\[
\frac{2x - 14}{x^2 - 49} + \frac{7x + 49}{x^2 - 49} = \frac{(2x - 14) + (7x + 49)}{x^2 - 49}
\][/tex]
5. Combine and simplify the numerator:
Combine like terms in the numerator:
[tex]\[
(2x - 14) + (7x + 49) = 2x + 7x - 14 + 49 = 9x + 35
\][/tex]
6. Write the final simplified expression:
[tex]\[
\frac{9x + 35}{x^2 - 49}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{2}{x+7}+\frac{7}{x-7}\)[/tex] is [tex]\(\frac{9x + 35}{x^2 - 49}\)[/tex]. The correct answer is:
A. [tex]\(\frac{9x + 35}{x^2 - 49}\)[/tex]
