To solve the quadratic equation \(\frac{(x+27)^2}{-6} = -3\), we start by eliminating the fraction. Multiply both sides by \(-6\):
\[ (x+27)^2 = -3 \times -6 \]
This simplifies to:
\[ (x+27)^2 = 18 \]
Next, we take the square root of both sides:
\[ x + 27 = \pm \sqrt{18} \]
Since \(\sqrt{18} = 3\sqrt{2}\), we can write:
\[ x + 27 = \pm 3\sqrt{2} \]
Now, solving for \(x\):
- For the positive case: \[ x = -27 + 3\sqrt{2} \]
- For the negative case: \[ x = -27 - 3\sqrt{2} \]
Next, we calculate the approximate values for \(x\):
-
Calculate \(3\sqrt{2}\): \[ \sqrt{2} \approx 1.414 \] \[ 3\sqrt{2} \approx 3 \times 1.414 \approx 4.242 \]
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For the positive case: \[ x \approx -27 + 4.242 \approx -22.758 \] Rounding to the nearest hundredth gives: \[ x \approx -22.76 \]
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For the negative case: \[ x \approx -27 - 4.242 \approx -31.242 \] Rounding to the nearest hundredth gives: \[ x \approx -31.24 \]
Thus, the solutions to the equation are:
\[ \boxed{-22.76} \quad \text{and} \quad \boxed{-31.24} \]