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 need to find the square roots of 2564.

Calculating the square roots:

1. The positive solution is \( x = \sqrt{2564} \).
2. The negative solution is \( x = -\sqrt{2564} \).

Now, we can simplify \( \sqrt{2564} \). First, let's factor 2564:

• 2564 is an even number, so we can divide by 2: \( 2564 \div 2 = 1282 \)
• Continuing to factor: \( 1282 \div 2 = 641 \)

Now, we check if 641 can be factored further. The prime factors of 641 are checked, and it turns out that 641 is a prime number.

So we can express \( 2564 \) as: \[ 2564 = 2^2 \times 641 \]

Now taking the square root: \[ \sqrt{2564} = \sqrt{2^2 \times 641} = 2\sqrt{641} \]

Thus, the solutions are:

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

Since the question asks for the lesser number first, the solutions in simplest fraction form are: \[ -\sqrt{2564}, \quad \sqrt{2564} \quad \text{or} \quad -2\sqrt{641}, \quad 2\sqrt{641}. \]

So the final answer is: \[ \boxed{-2\sqrt{641}, 2\sqrt{641}}. \]