To solve the quadratic equation \(\frac{(x+16)^2}{9} = 7\), we will first eliminate the fraction by multiplying both sides by 9:
\[ (x+16)^2 = 63 \]
Next, we take the square root of both sides. Remember to consider both the positive and negative roots:
\[ x + 16 = \pm \sqrt{63} \]
Calculating \(\sqrt{63}\):
\[ \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \approx 3 \cdot 2.64575 \approx 7.937 \]
Now we substitute this back into our equation:
\[ x + 16 = 3\sqrt{7} \quad \text{or} \quad x + 16 = -3\sqrt{7} \]
Now we solve for \(x\) in each case.
- For \(x + 16 = 3\sqrt{7}\):
\[ x = 3\sqrt{7} - 16 \approx 7.937 - 16 \approx -8.063 \]
- For \(x + 16 = -3\sqrt{7}\):
\[ x = -3\sqrt{7} - 16 \approx -7.937 - 16 \approx -23.937 \]
Thus the solutions to the equation \(\frac{(x+16)^2}{9} = 7\) rounded to the nearest hundredth are:
\[ x \approx -8.06 \quad \text{and} \quad x \approx -23.94 \]
So the final answers are:
\[ \boxed{-8.06} \quad \text{and} \quad \boxed{-23.94} \]
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