To solve the equation \( x^2 = 2564 \) by inspection, we need to find the square roots of 2564.
The two real solutions for this equation are given by:
- \( x = \sqrt{2564} \)
- \( x = -\sqrt{2564} \)
Calculating \( \sqrt{2564} \):
First, we can try to simplify the square root to find the simplest fraction:
\[ 2564 = 4 \times 641 \]
Taking the square root:
\[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]
Thus, the two solutions are:
- \( x = -2\sqrt{641} \)
- \( x = 2\sqrt{641} \)
Since we are instructed to enter the lesser number first, the answers are:
The first solution is \( x = -2\sqrt{641} \).
The second solution is \( x = 2\sqrt{641} \).