We start with the equation given:
\[ (x + 6)^2 = 8 \]
To solve for \(x\), we first take the square root of both sides:
\[ x + 6 = \pm \sqrt{8} \]
Calculating \(\sqrt{8}\):
\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
Now we have two equations to solve:
- \(x + 6 = 2\sqrt{2}\)
- \(x + 6 = -2\sqrt{2}\)
Solving the first equation:
\[ x + 6 = 2\sqrt{2} \] \[ x = 2\sqrt{2} - 6 \]
Solving the second equation:
\[ x + 6 = -2\sqrt{2} \] \[ x = -2\sqrt{2} - 6 \]
Now let’s compute the numerical values of these solutions.
- For \(x = 2\sqrt{2} - 6\):
Calculating \(2\sqrt{2}\): \(\sqrt{2} \approx 1.414\) \[ 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \] Thus: \[ x \approx 2.828 - 6 \approx -3.172 \]
- For \(x = -2\sqrt{2} - 6\): \[ -2\sqrt{2} \approx -2.828 \] Thus: \[ x \approx -2.828 - 6 \approx -8.828 \]
Now we approximate to check against the options given:
- \(-3.172\) and \(-8.828\) do not match any of the provided numerical solutions exactly.
To match the format of the responses, let’s analyze if there are any recognizable patterns in the given response options after recalculating:
Using a calculator for more precise values could help:
- \(2\sqrt{2} \approx 2.828\) makes \(x \approx -3.172\)
- \(-2\sqrt{2} \approx -2.828\) makes \(x \approx -8.828\)
None of the given sets, such as {-0.34, -11.66}, {-1.17, -6.83}, {-2, -10}, {10, -22}, match our results.
Considering that our approximations are not matching approximations of any of the given choices, it might require verification with explicit selections to check which is closest.
To conclude: The above calculation did not yield exact matching values. If we examine the roots with potential variable tracking errors or systematic checks through the context, the most suitable answer, given the repetition to clarify, would be to retrieve more specific instruction context or numerical reconciliation if aids are present for numeric errors.
None truly fits; however, best approximative checks or context for information can direct multiple choices usability!