To solve the equation \( x^2 = 1625 \) by inspection, we want to find the square roots of 1625.
First, we can estimate \( x \) by determining the perfect squares around 1625.
We know:
- \( 40^2 = 1600 \)
- \( 41^2 = 1681 \)
So, \( x^2 = 1625 \) lies between \( 40^2 \) and \( 41^2 \). Thus, we can hypothesize that \( x \) is between 40 and 41.
For a more precise inspection, we can see if we can simplify \( \sqrt{1625} \):
\[ 1625 = 25 \times 65 = 25 \times (5 \times 13) \]
This allows us to take the square root:
\[ \sqrt{1625} = \sqrt{25 \times 65} = \sqrt{25} \times \sqrt{65} = 5\sqrt{65} \]
Now, we have two real solutions:
\[ x = 5\sqrt{65} \quad \text{and} \quad x = -5\sqrt{65} \]
At this point, we can approximate \( \sqrt{65} \). Since \( 8^2 = 64 \) and \( 9^2 = 81 \), we know \( \sqrt{65} \) is a bit more than 8.
Calculating further, we find:
\[ \sqrt{65} \approx 8.06 \]
Thus,
\[ 5\sqrt{65} \approx 5 \times 8.06 \approx 40.3 \]
So, the two real solutions are approximately:
\[ x \approx 40.3 \quad \text{and} \quad x \approx -40.3 \]
In conclusion, the two real solutions to the equation \( x^2 = 1625 \) are:
\[ x = 5\sqrt{65} \quad \text{and} \quad x = -5\sqrt{65} \]
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