To solve the quadratic equation \((x + 6)^2 = 8\), we can take the following steps:
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Take the square root of both sides: \[ x + 6 = \pm \sqrt{8} \]
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Simplify \(\sqrt{8}\): \[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]
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Now, rewrite the equation: \[ x + 6 = 2\sqrt{2} \quad \text{or} \quad x + 6 = -2\sqrt{2} \]
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Solve for \(x\) in both cases:
For \(x + 6 = 2\sqrt{2}\): \[ x = 2\sqrt{2} - 6 \]
For \(x + 6 = -2\sqrt{2}\): \[ x = -2\sqrt{2} - 6 \]
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Now we will calculate the numerical values:
\(\sqrt{2} \approx 1.414\), so:
- \(2\sqrt{2} \approx 2 \times 1.414 \approx 2.828\)
- Therefore, \( x = 2.828 - 6 \approx -3.172 \)
- \( x = -2.828 - 6 \approx -8.828 \)
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Thus, the approximate solutions are: \[ { -3.172, -8.828 } \]
The given options do not match precisely with these calculations. We can look at the approximate decimal values and check the options provided:
- {10, −22}
- {−1.17, −6.83}
- {−2, −10}
- {−0.34, −11.66}
None of the provided responses are correct based on the calculations.
Please double-check your equation, or if "8" in the equation might have been another value. Otherwise, the correct process yields approximate solutions \(-3.172\) and \(-8.828\).