To solve the equation \( x^2 = 2564 \) by inspection, we need to find the values of \( x \) that satisfy this equation.
Taking the square root of both sides gives us: \[ x = \sqrt{2564} \quad \text{or} \quad x = -\sqrt{2564} \]
Calculating \( \sqrt{2564} \):
- First, let's simplify \( 2564 \). We can find its prime factorization.
- \( 2564 \) is even, so we can divide it by \( 2 \): \[ 2564 \div 2 = 1282 \]
- Continuing to factor \( 1282 \): \[ 1282 \div 2 = 641 \]
- Now, \( 641 \) is a prime number. Thus, the prime factorization of \( 2564 \) is: \[ 2^2 \times 641 \]
Using this, we can find the square root: \[ \sqrt{2564} = \sqrt{2^2 \times 641} = 2\sqrt{641} \]
Thus the two solutions to the equation \( x^2 = 2564 \) are: \[ x = 2\sqrt{641} \quad \text{and} \quad x = -2\sqrt{641} \]
Since the problem asks for the lesser number first, our final answer in simplest fractional form will be: \[ {-2\sqrt{641}, 2\sqrt{641}} \]