To solve the equation \( x^2 = 81 \) by inspection, we can take the square root of both sides.
We find:
\[ x = \sqrt{81} \quad \text{or} \quad x = -\sqrt{81} \]
Calculating the square root:
\[ \sqrt{81} = 9 \]
Thus, we have two potential solutions:
\[ x = 9 \quad \text{and} \quad x = -9 \]
Therefore, the complete solution set is:
\[ { -9, 9 } \]
To summarize, the solutions to the equation \( x^2 = 81 \) are \( -9 \) and \( 9 \). There are no other real solutions.