To solve the quadratic equation \(\frac{(x+16)^2}{9} = 7\), we will first isolate the squared term by multiplying both sides by 9:
\[ (x + 16)^2 = 7 \times 9 \]
\[ (x + 16)^2 = 63 \]
Next, we take the square root of both sides. Remember to consider both the positive and negative square roots:
\[ x + 16 = \pm \sqrt{63} \]
Now, simplify \(\sqrt{63}\):
\[ \sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7} \]
So we have:
\[ x + 16 = 3\sqrt{7} \quad \text{or} \quad x + 16 = -3\sqrt{7} \]
Now, we can solve for \(x\) in both cases:
- \(x + 16 = 3\sqrt{7}\)
\[ x = 3\sqrt{7} - 16 \]
- \(x + 16 = -3\sqrt{7}\)
\[ x = -3\sqrt{7} - 16 \]
Next, we will calculate the approximate values of \(3\sqrt{7}\) and \(-3\sqrt{7}\) to find the values of \(x\).
First, let's calculate \(3\sqrt{7}\):
\[ \sqrt{7} \approx 2.6457513110645906 \quad \Rightarrow \quad 3\sqrt{7} \approx 3 \times 2.6457513110645906 \approx 7.9372539331 \]
So,
\[ x \approx 7.9372539331 - 16 \approx -8.0627460669 \quad \Rightarrow \quad x \approx -8.06 \quad (\text{rounded to the nearest hundredth}) \]
Now for the second case:
\[ -3\sqrt{7} \approx -7.9372539331 \quad \Rightarrow \quad x \approx -7.9372539331 - 16 \approx -23.9372539331 \quad \Rightarrow \quad x \approx -23.94 \quad (\text{rounded to the nearest hundredth}) \]
Thus, the two solutions for the equation \(\frac{(x+16)^2}{9} = 7\) are:
\[ \boxed{-8.06} \quad \text{and} \quad \boxed{-23.94} \]
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