To solve the equation \( x^2 = 1625 \) by inspection, we want to find the square roots of \( 1625 \).
Calculating the square root, we have:
\[ x = \pm \sqrt{1625} \]
To simplify \( \sqrt{1625} \):
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Factor \( 1625 \): \[ 1625 = 25 \times 65 = 25 \times (5 \times 13) = 5^2 \times 5 \times 13 \]
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Thus, \[ \sqrt{1625} = \sqrt{5^2 \times 5 \times 13}= 5\sqrt{65} \]
The value of \( x \) can therefore be expressed as:
\[ x = \pm 5\sqrt{65} \]
Thus, the two real solutions are:
\[ -5\sqrt{65} \quad \text{and} \quad 5\sqrt{65} \]
In simplest fraction form, the solutions are:
\[ \boxed{-5\sqrt{65}} \quad \text{and} \quad \boxed{5\sqrt{65}} \]