To solve the equation \( x^2 = 2564 \) by inspection, we can determine the two solutions by taking the square root of both sides.
The solutions are given by:
\[ x = \pm \sqrt{2564} \]
Now, let's calculate the square root:
To find \( \sqrt{2564} \), we can factor \( 2564 \) to see if it can be simplified further:
\[ 2564 = 4 \times 641 \]
Thus:
\[ \sqrt{2564} = \sqrt{4 \times 641} = \sqrt{4} \times \sqrt{641} = 2\sqrt{641} \]
Now, the two solutions to the equation \( x^2 = 2564 \) are:
\[ x = 2\sqrt{641} \quad \text{and} \quad x = -2\sqrt{641} \]
To provide the answers in the specified format, the lesser number first:
The solutions in simplest fraction form are:
\[ -2\sqrt{641}, , 2\sqrt{641} \]
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