To solve for $x$ when given the equation:

$x^2 = \frac{81}{144}$

we start by taking the square root of both sides:

$x = \pm \sqrt{\frac{81}{144}}$

Now, we compute the square root of the fraction by taking the square root of the numerator and the denominator separately:

$x = \pm \frac{\sqrt{81}}{\sqrt{144}}$

Since $\sqrt{81} = 9$ and $\sqrt{144} = 12$, we get:

$x = \pm \frac{9}{12}$

Now, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$x = \pm \frac{9 \div 3}{12 \div 3} = \pm \frac{3}{4}$

Thus, the solution for $x$ is:

$x = \frac{3}{4} \quad \text{or} \quad x = -\frac{3}{4}$

Would you like to see more details or have any questions about this solution?

Related Questions:

1. What are the steps to solve quadratic equations in general?
2. How do you find the square root of fractions?
3. What are the rules for simplifying fractions?
4. Can you solve for $x$ if the equation was $x^2 = 144/81$?
5. How do you solve for $x$ when given a cubic equation?

Tip: When solving equations involving square roots, remember to consider both the positive and negative roots.