To solve the equation \( x^2 = \frac{36}{169} \), we can take the square root of both sides.
The square root of both sides gives us:
\[ x = \pm \sqrt{\frac{36}{169}} \]
Since \( \sqrt{\frac{36}{169}} = \frac{\sqrt{36}}{\sqrt{169}} = \frac{6}{13} \), we have:
\[ x = \pm \frac{6}{13} \]
This results in two possible solutions:
\[ x = \frac{6}{13} \quad \text{and} \quad x = -\frac{6}{13} \]
Thus, the complete solution set is:
\[ { -\frac{6}{13}, \frac{6}{13} } \]
So the correct answer is:
D: \( { -\frac{6}{13}, \frac{6}{13} } \)