To solve the equation \( x^2 = 1625 \) by inspection, we can take the square root of both sides. The solutions will be:
\[ x = \sqrt{1625} \quad \text{and} \quad x = -\sqrt{1625} \]
Next, we simplify \( \sqrt{1625} \) for easier comparison. We can factor \( 1625 \) to find its prime factors.
\[ 1625 = 25 \times 65 = 25 \times (5 \times 13) = 5^2 \times 5 \times 13 = 5^3 \times 13 \]
Thus, we can write:
\[ \sqrt{1625} = \sqrt{25 \times 65} = \sqrt{25} \times \sqrt{65} = 5\sqrt{65} \]
Therefore, the two solutions are:
\[ x = -5\sqrt{65} \quad \text{and} \quad x = 5\sqrt{65} \]
Next, we express the solutions in simplest fractional form. The solutions are:
\[ x = -5\sqrt{65} \quad \text{and} \quad x = 5\sqrt{65} \]
Thus, the final answer in the required format is:
\[ {-5\sqrt{65}, 5\sqrt{65}} \]
If you need the decimal approximations, \(\sqrt{65} \approx 8.062\), and thus:
\[ x \approx -5 \times 8.062 \approx -40.31 \quad \text{and} \quad x \approx 5 \times 8.062 \approx 40.31 \]
But in simplest radical form, the final answer is:
\[ {-5\sqrt{65}, 5\sqrt{65}} \]
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