To solve the equation \((x + 6)^2 = 8\) using square roots, follow these steps:
- Take the square root of both sides:
\[ x + 6 = \pm \sqrt{8} \]
- Simplify \(\sqrt{8}\):
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
So, we have:
\[ x + 6 = \pm 2\sqrt{2} \]
- Solve for \(x\):
\[ x = -6 \pm 2\sqrt{2} \]
Now we will find the numerical values:
- For the positive case:
\[ x = -6 + 2\sqrt{2} \approx -6 + 2(1.414) \approx -6 + 2.828 \approx -3.172 \]
- For the negative case:
\[ x = -6 - 2\sqrt{2} \approx -6 - 2(1.414) \approx -6 - 2.828 \approx -8.828 \]
Thus, the approximate solutions are:
- \(x \approx -3.17\)
- \(x \approx -8.83\)
Since the options provided don’t exactly match this, let’s examine the closest values:
\(-2\) could be obtained if we miscalculated \(\sqrt{2}\) in some way. But if we focus on the calculations, the roots actually do not match any of the given options closely.
The solutions calculated were estimated and did not yield one of the provided choices directly. Therefore, I would recommend carefully rechecking the working or considering if there might have been an error in interpreting the possible choices.
Given the problem statement and an estimate around where they fall numerically, the answer would be:
None of the above options seem correct.
The calculations reveal solutions \({-6 - 2\sqrt{2}, -6 + 2\sqrt{2}}\). If checked against numerical computations they do not perfectly align to the fragments you presented.)
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