To solve the quadratic equation \(\frac{(x + 27)^2}{-6} = -3\), we can begin by eliminating the fraction.
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Multiply both sides by -6 to eliminate the denominator: \[ (x + 27)^2 = -3 \times -6 \] \[ (x + 27)^2 = 18 \]
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Next, we take the square root of both sides: \[ x + 27 = \pm \sqrt{18} \] \[ x + 27 = \pm 3\sqrt{2} \]
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To solve for \(x\), we subtract 27 from both sides: \[ x = -27 \pm 3\sqrt{2} \]
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This gives us two possible solutions: \[ x_1 = -27 + 3\sqrt{2} \] \[ x_2 = -27 - 3\sqrt{2} \]
Now, let's calculate the values of \(x_1\) and \(x_2\).
First, we compute \(3\sqrt{2}\): \[ \sqrt{2} \approx 1.414 \] \[ 3\sqrt{2} \approx 3 \times 1.414 \approx 4.242 \]
Now, we can find the two values:
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For \(x_1\): \[ x_1 = -27 + 4.242 \approx -22.758 \]
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For \(x_2\): \[ x_2 = -27 - 4.242 \approx -31.242 \]
Finally, rounding to the nearest hundredth: \[ x_1 \approx -22.76 \] \[ x_2 \approx -31.24 \]
The smaller value is \(x_2\). Therefore, the two solutions are: \[ \boxed{-31.24} \] \[ \boxed{-22.76} \]