To solve the equation \( x^2 = 2564 \) by inspection, we first take the square root of both sides:
\[ x = \pm \sqrt{2564} \]
Next, we need to simplify \( \sqrt{2564} \). We will first factor \( 2564 \):
- Check if \( 2564 \) is even, which it is. Divide by \( 2 \): \[ 2564 \div 2 = 1282 \]
- \( 1282 \) is also even. Divide by \( 2 \) again: \[ 1282 \div 2 = 641 \]
- Now we need to determine if \( 641 \) is prime. We can check for divisibility by prime numbers up to \( \sqrt{641} \) (which is approximately \( 25.3 \)). \( 641 \) is not divisible by \( 2, 3, 5, 7, 11, 13, 17, 19, \) or \( 23 \), indicating that \( 641 \) is indeed a prime number.
So we have the prime factorization: \[ 2564 = 2^2 \times 641 \]
Now we can compute the square root: \[ \sqrt{2564} = \sqrt{2^2 \times 641} = 2\sqrt{641} \]
Therefore, the solutions to the equation \( x^2 = 2564 \) are: \[ x = 2\sqrt{641} \quad \text{and} \quad x = -2\sqrt{641} \]
Since we need to provide the lesser solution first, we will express the solutions in simplest fraction form. Here, they are already in simplest form.
Thus, the solutions are:
\[ {-2\sqrt{641}, 2\sqrt{641}} \]
If you need numerical approximations, \( \sqrt{641} \) is approximately \( 25.31 \), making: \[ 2\sqrt{641} \approx 50.62 \quad \text{and} \quad -2\sqrt{641} \approx -50.62 \]
In the simplest fraction form, the solutions are:
\[ {-2\sqrt{641}, 2\sqrt{641}} \]