To solve the equation \( x^2 = 1010 \), we need to find the values of \( x \) such that when squared they equal 1010.
First, we can take the square root of both sides:
\[ x = \sqrt{1010} \quad \text{or} \quad x = -\sqrt{1010} \]
Next, we can calculate \( \sqrt{1010} \). We know that \( 31^2 = 961 \) and \( 32^2 = 1024 \), so \( \sqrt{1010} \) is between 31 and 32.
Calculating it approximately, we find:
\[ \sqrt{1010} \approx 31.78 \]
Therefore, the two solutions are:
\[ x \approx 31.78 \quad \text{and} \quad x \approx -31.78 \]
Now, we can compare this result with the provided options:
- \( {105} \)
- \( {10^5} \)
- \( {-105, 105} \)
- \( {-5^5} \)
- \( {-510, 510} \)
None of these options directly match our approximated values of \( \sqrt{1010} \) and \( -\sqrt{1010} \), which are approximately 31.78 and -31.78.
Since 31.78 does not match any values among the provided options, it seems there is a possible mistake in the options listed. If the options were meant to be in the context of whole numbers or specific roots, they do not accurately reflect the calculated roots of \( x^2 = 1010 \).
Thus, the correct solution set based on our calculations should be:
\[ {-\sqrt{1010}, \sqrt{1010}} \text{ or approximately } {-31.78, 31.78} \]
But given the provided responses, none are correct. If adjusted, the closest choices to the characteristic structure could imply \( {-105, 105} \) as basic solutions for a similar equation, but they do not relate to the accurate results obtained here.
In summary, none of the provided choices are correct if we are strictly considering the original equation \( x^2 = 1010 \).
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