To solve the equation \( x^2 = 2564 \) by inspection, we take the square root of both sides:
\[ x = \pm \sqrt{2564} \]
Now, we need to find \( \sqrt{2564} \). First, we can factor 2564 to make it simpler to find the square root.
We can check if 2564 is divisible by 4 (since the last two digits, 64, are divisible by 4):
\[ 2564 \div 4 = 641 \]
Next, we check if 641 is prime. After checking divisibility with prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, etc.), we find that 641 is not divisible by any of these, meaning 641 is prime.
So we have the prime factorization:
\[ 2564 = 4 \times 641 = 2^2 \times 641 \]
Now we can take the square root:
\[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]
Thus, the two solutions for \( x \) are:
\[ x = 2\sqrt{641} \quad \text{and} \quad x = -2\sqrt{641} \]
Putting these into the proper format you requested, with the lesser solution first:
The first solution is: \[ x = -2\sqrt{641} \]
The second solution is: \[ x = 2\sqrt{641} \]