To solve the equation \( x^2 = 1625 \) by inspection, we can find the square roots of 1625.
First, let's simplify 1625: \[ 1625 = 25 \times 65 = 25 \times (5 \times 13) = 5^2 \times 5 \times 13 = 5^3 \times 13 \]
Now, we can find the square root: \[ \sqrt{1625} = \sqrt{25 \times 65} = \sqrt{25} \cdot \sqrt{65} = 5\sqrt{65} \]
This means the solutions to the equation \( x^2 = 1625 \) are: \[ x = 5\sqrt{65} \quad \text{and} \quad x = -5\sqrt{65} \]
To express these as fractions, we can write: \[ 5\sqrt{65} = \frac{5\sqrt{65}}{1} \quad \text{and} \quad -5\sqrt{65} = \frac{-5\sqrt{65}}{1} \]
So the two real solutions, in simplest fraction form and with the lesser number first, are: \[ \boxed{\left(-5\sqrt{65}, 5\sqrt{65}\right)} \]
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