To solve the equation \( (x + 6)^2 = 8 \) using square roots, we can follow these steps:
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{8} \]
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Simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \] So, we have: \[ x + 6 = \pm 2\sqrt{2} \]
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Now, we can solve for \(x\) by isolating it: \[ x = -6 \pm 2\sqrt{2} \]
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This gives us two potential solutions: \[ x = -6 + 2\sqrt{2} \quad \text{and} \quad x = -6 - 2\sqrt{2} \]
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Now, we can calculate the numerical values.
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For \( x = -6 + 2\sqrt{2} \): \[ \approx -6 + 2 \times 1.414 \approx -6 + 2.828 = -3.172 \]
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For \( x = -6 - 2\sqrt{2} \): \[ \approx -6 - 2 \times 1.414 \approx -6 - 2.828 = -8.828 \]
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Therefore, the solutions in decimal form are approximately: \[ x \approx -3.17 \quad \text{and} \quad x \approx -8.83 \]
Comparing the approximate values to the given responses, none of the options match the values calculated above. So, let's confirm the calculations.
Lastly, simplifying:
- The negative solution \( -6 - 2\sqrt{2} \) is approximately \( -8.83 \).
- The positive solution \( -6 + 2\sqrt{2} \) is approximately \( -3.17 \).
The correct approximate values \( {-3.17, -8.83} \) do not appear in the response options provided, implying that the question options may have different representations or calculations that require double-checking.
Please validate the responses against the original question's constraints or clarify if the responses listed are meant to include different reference points or approximations.