To solve the equation \( x^2 = 2564 \) by inspection, we can take the square root of both sides. This gives us:
\[ x = \sqrt{2564} \quad \text{or} \quad x = -\sqrt{2564} \]
First, we need to simplify \( \sqrt{2564} \).
To do this, we can factor 2564:
\[ 2564 = 4 \times 641 \]
Taking the square root gives us:
\[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]
Thus, the two solutions for \( x \) are:
\[ x = 2\sqrt{641} \quad \text{and} \quad x = -2\sqrt{641} \]
Now we write the solutions in simplest fraction form:
\[ {-2\sqrt{641}, 2\sqrt{641}} \]
The lesser solution is listed first.