To solve the equation \(x^2 = 1625\) by inspection, we can take the square root of both sides.
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The equation implies that: \[ x = \sqrt{1625} \quad \text{or} \quad x = -\sqrt{1625} \]
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First, we simplify \(\sqrt{1625}\):
- We can factor \(1625\) as follows: \[ 1625 = 25 \times 65 = 25 \times (5 \times 13) = 5^2 \times 5 \times 13 \]
- Therefore, we can find the square root: \[ \sqrt{1625} = \sqrt{25 \times 65} = \sqrt{25} \times \sqrt{65} = 5\sqrt{65} \]
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The two solutions to the equation \(x^2 = 1625\) are: \[ x = 5\sqrt{65} \quad \text{and} \quad x = -5\sqrt{65} \]
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Since we need to present the lesser number first, the solutions are: \[ {-5\sqrt{65}, 5\sqrt{65}} \]
Thus, the final answer, in the simplest fraction form, is: \[ {-5\sqrt{65}, 5\sqrt{65}} \]