To solve the equation \( x^2 = 1010 \) by inspection, you can find the square roots of 1010.
Calculating the square roots, you get:
\[ x = \sqrt{1010} \quad \text{or} \quad x = -\sqrt{1010} \]
Next, we can estimate \(\sqrt{1010}\). Since \(31^2 = 961\) and \(32^2 = 1024\), we can determine that \(\sqrt{1010}\) is between 31 and 32.
It's a little more than 31, but definitely less than 32. However, the specific values provided in your options do not seem to align with \(\sqrt{1010}\):
- \(-105, 105\) corresponds to \(-105^2 = 11025\)
- \(105\) corresponds to \(105^2 = 11025\)
- \(-55\) corresponds to \(-5^2 = 25\)
- \(-510, 510\) corresponds to \(-510^2 = 260100\)
None of these options seem to accurately represent the solution to \( x^2 = 1010 \).
If we needed to choose from the given options, they all provide incorrect squared values relative to 1010.
However, if you're looking for the square roots more directly relevant to 1010, we have: \({ -\sqrt{1010}, \sqrt{1010} }\)
None of the provided sets seems correct based on this estimation. The closest correct understanding is that the solution set is approximately \({ -31.78, 31.78 }\), which does not correspond to the available options.
If you must select the closest option that fits the ± notation, you might choose \({-105, 105}\) for its representation of dual roots, but remember these values are not correct in terms of solving \(x^2 = 1010\). The answer you should provide based on accuracy is that none of these options accurately solve the equation given.
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