To solve the quadratic equation \(\frac{(x + 27)^2}{-6} = -3\), we start by eliminating the fraction. Multiply both sides by \(-6\):
\[ (x + 27)^2 = -3 \cdot -6 \] \[ (x + 27)^2 = 18 \]
Next, we take the square root of both sides. Remember to consider both the positive and negative square roots:
\[ x + 27 = \sqrt{18} \quad \text{or} \quad x + 27 = -\sqrt{18} \]
Now, simplify \(\sqrt{18}\):
\[ \sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2} \]
So, we have:
\[ x + 27 = 3\sqrt{2} \quad \text{or} \quad x + 27 = -3\sqrt{2} \]
Now, solve for \(x\) in both cases:
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First case: \[ x = 3\sqrt{2} - 27 \]
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Second case: \[ x = -3\sqrt{2} - 27 \]
Now, calculate \(3\sqrt{2}\) (using \(\sqrt{2} \approx 1.414\)):
\[ 3\sqrt{2} \approx 3 \cdot 1.414 \approx 4.242 \]
Therefore, for the first case:
\[ x \approx 4.242 - 27 \approx -22.758 \]
For the second case:
\[ -3\sqrt{2} \approx -4.242 \] \[ x \approx -4.242 - 27 \approx -31.242 \]
Thus, the two solutions are:
\[ x \approx -22.76 \quad \text{(rounded to the nearest hundredth)} \] \[ x \approx -31.24 \quad \text{(rounded to the nearest hundredth)} \]
Final Solutions: \[ x \approx -22.76 \quad \text{and} \quad x \approx -31.24 \]