To solve the quadratic equation \((x + 27)^2 / -6 = -3\), let's start by isolating the squared term:
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Multiply both sides by -6 to eliminate the fraction: \[ (x + 27)^2 = -3 \times -6 \] \[ (x + 27)^2 = 18 \]
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Now take the square root of both sides: \[ x + 27 = \pm \sqrt{18} \] Simplifying \(\sqrt{18}\): \[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \] (Approximately \(3 \times 1.414 \approx 4.242\))
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Now, we have: \[ x + 27 = 3\sqrt{2} \quad \text{or} \quad x + 27 = -3\sqrt{2} \]
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Solve for \(x\) by subtracting 27 from both sides in both cases: \[ x = -27 + 3\sqrt{2} \quad \text{or} \quad x = -27 - 3\sqrt{2} \]
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Calculate \(x\): For \(x = -27 + 3\sqrt{2}\): \[ x \approx -27 + 4.242 \approx -22.76 \] For \(x = -27 - 3\sqrt{2}\): \[ x \approx -27 - 4.242 \approx -31.24 \]
So the correct answers are approximately \(-22.76\) and \(-31.24\).
Why is \(-23\) the wrong answer?
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\(-23\) is incorrect because it does not satisfy the original equation. When substituting \(-23\) back in:
\[ (x + 27)^2 / -6 = -3 \ (-23 + 27)^2 / -6 = -3 \ (4)^2 / -6 \neq -3 \quad \text{(This simplifies to } \frac{16}{-6} = -\frac{8}{3}, \text{ which is not } -3) \]
Thus, \(-23\) is not a valid solution of the equation.
In conclusion, the correct solutions, rounded to two decimal places, are indeed approximately \(-22.76\) and \(-31.24\).