Question

Solve x2=2564.
There are two real solutions. Enter the lesser number first.
Leave the answers in simplest fraction form

Answer 1

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:

\[
\boxed{-2\sqrt{641}} \quad \text{and} \quad \boxed{2\sqrt{641}}
\]