Solve x2=2564 by inspection. There are two real solutions. Enter the lesser number first. Leave the answers in simplest fraction form. (1 point)

Answer 1

To solve the equation \( x^2 = 2564 \) by inspection, we can take the square root of both sides.

Taking the square root gives us two potential solutions:

\[ x = \sqrt{2564} \quad \text{and} \quad x = -\sqrt{2564} \]

Next, we need to simplify \( \sqrt{2564} \). First, we can find the prime factorization or identify a perfect square factor of 2564.

Let's start with dividing by 4 since 2564 is even:

\[ 2564 \div 4 = 641 \]

Now, we know:

\[ 2564 = 4 \times 641 \]

Taking the square root gives us:

\[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]

So our solutions are:

1. \( x = 2\sqrt{641} \)
2. \( x = -2\sqrt{641} \)

Now, we want to express the solutions in simplest fraction form. Since \( \sqrt{641} \) cannot be simplified further (641 is a prime number), we have:

The lesser number first is:

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

Thus, the final answer is:

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