To solve the equation \( x^2 = 2564 \) by inspection, we need to find two numbers that, when squared, equal 2564.
Taking the square root of both sides:
\[ x = \pm \sqrt{2564} \]
Next, we can simplify \( \sqrt{2564} \). First, let's find the prime factorization of 2564.
- 2564 is even, so we can divide by 2: \[ 2564 \div 2 = 1282 \]
- 1282 is also even: \[ 1282 \div 2 = 641 \]
- Now, we check if 641 is a prime number. It is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or even up to its approximate square root (around 25). Therefore, 641 is a prime number.
So, the prime factorization of 2564 is: \[ 2564 = 2^2 \times 641 \]
Now we can find the square root: \[ \sqrt{2564} = \sqrt{2^2 \times 641} = 2 \sqrt{641} \]
Thus, the solutions for \( x \) are: \[ x = \pm 2\sqrt{641} \]
The lesser of the two solutions is: \[ -2\sqrt{641} \]
And the greater solution is: \[ 2\sqrt{641} \]
Putting the answers in simplest fraction form, we have: \[ { -2\sqrt{641}, 2\sqrt{641} } \]
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