To solve the equation \((x + 6)^{2} = 8\), we start by taking the square root of both sides. Here are the steps:
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{8} \]
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Simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \] Therefore, we have: \[ x + 6 = \pm 2\sqrt{2} \]
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Now, we will isolate \(x\) by subtracting 6 from both sides: \[ x = -6 \pm 2\sqrt{2} \]
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This gives us two solutions: \[ x = -6 + 2\sqrt{2} \] \[ x = -6 - 2\sqrt{2} \]
Now we will approximate these values to determine which of the given options they match.
Calculating \(2\sqrt{2}\): \[ \sqrt{2} \approx 1.414 \implies 2\sqrt{2} \approx 2 \cdot 1.414 \approx 2.828 \]
So, we calculate each solution:
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For \(x = -6 + 2\sqrt{2}\): \[ x \approx -6 + 2.828 \approx -3.172 \]
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For \(x = -6 - 2\sqrt{2}\): \[ x \approx -6 - 2.828 \approx -8.828 \]
So our approximate solutions are: \[ x \approx -3.17 \quad \text{and} \quad x \approx -8.83 \]
Now let's check if these results correspond with the given options:
- \({10,−22}\)
- \({-2,−10}\)
- \({-0.34,−11.66}\)
- \({-1.17,−6.83}\)
None of these options seem to match the calculations exactly. However, if we check the last option and do a deeper verification, we would need to reconsider \(-6\) and the magnitude of \(2\sqrt{2}\):
- For \(-6\), we found approximate values that don't match.
In conclusion, I derived that the roots don't correspond exactly with the choices you've given. Please make sure to recheck the question or any numerical solutions if there were specific roots expected based on different manipulation of the equation.