To solve the equation \(x^2 = 2564\), we need to take the square root of both sides.
\[ x = \pm \sqrt{2564} \]
Now, we need to simplify \(\sqrt{2564}\). First, we can factor 2564 into its prime factors.
We start by checking if it is even:
\[ 2564 \div 2 = 1282 \] \[ 1282 \div 2 = 641 \]
Now, we check if 641 is a prime number. It is not divisible by any prime numbers up to its square root (\(\sqrt{641} \approx 25.3\)). Checking divisibility:
- Not divisible by 3 (sum of digits is 11).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by 7 (641 ÷ 7 = 91.57).
- Not divisible by 11 (641 ÷ 11 = 58.27).
- Not divisible by 13 (641 ÷ 13 = 49.31).
- Not divisible by 17 (641 ÷ 17 = 37.12).
- Not divisible by 19 (641 ÷ 19 = 33.74).
- Not divisible by 23 (641 ÷ 23 = 27.87).
Since 641 is not divisible by any primes up to its square root, we conclude that it is prime.
Thus, the complete factorization of 2564 is:
\[ 2564 = 2^2 \times 641 \]
Now we can simplify \(\sqrt{2564}\):
\[ \sqrt{2564} = \sqrt{2^2 \times 641} = 2\sqrt{641} \]
Now we can find the two solutions for \(x\):
\[ x = \pm 2\sqrt{641} \]
The two real solutions are:
\[ x_1 = -2\sqrt{641} \quad \text{(lesser solution)} \] \[ x_2 = 2\sqrt{641} \quad \text{(greater solution)} \]
Thus, the final solutions in simplest form are:
\[ \boxed{-2\sqrt{641}} \quad \text{and} \quad \boxed{2\sqrt{641}} \]