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 begin by taking the square root of both sides.

\[
x = \pm \sqrt{2564}
\]

Now we need to simplify \( \sqrt{2564} \). We can start by factoring 2564.

First, we check for divisibility by 2:

\[
2564 \div 2 = 1282
\]
\[
1282 \div 2 = 641
\]

Now, 641 is not divisible by 2, 3, 5, or 7, and checking further, we find that 641 is a prime number.

Thus, we can write the factorization of 2564 as:

\[
2564 = 2^2 \times 641
\]

Now we can find \( \sqrt{2564} \):

\[
\sqrt{2564} = \sqrt{2^2 \times 641} = 2\sqrt{641}
\]

Thus, the solutions for \( x \) are:

\[
x = \pm 2\sqrt{641}
\]

Now we have two real solutions, \( 2\sqrt{641} \) and \( -2\sqrt{641} \).

Since we need to enter the lesser number first, the two solutions will be:

\[
-2\sqrt{641}, \quad 2\sqrt{641}
\]

In simplest fraction form, we can express this as:

\[
-2\sqrt{641}, \quad 2\sqrt{641}
\]

These are the two real solutions.