To solve the equation \(\frac{(x + 16)^2}{9} = 7\), we start by eliminating the fraction. We can do this by multiplying both sides of the equation by 9:
\[ (x + 16)^2 = 63 \]
Next, we want to take the square root of both sides. Remember, when we take the square root, we must consider both the positive and negative roots:
\[ x + 16 = \pm \sqrt{63} \]
Calculating \(\sqrt{63}\):
\[ \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \cdot \sqrt{7} = 3\sqrt{7} \]
Now substituting back, we have:
\[ x + 16 = 3\sqrt{7} \quad \text{or} \quad x + 16 = -3\sqrt{7} \]
Next, we isolate \(x\):
-
For the positive case: \[ x = 3\sqrt{7} - 16 \]
-
For the negative case: \[ x = -3\sqrt{7} - 16 \]
Now we will compute the numerical values:
First, we need the approximate value of \(\sqrt{7}\):
\[ \sqrt{7} \approx 2.64575 \]
Calculating \(3\sqrt{7}\):
\[ 3\sqrt{7} \approx 3 \times 2.64575 \approx 7.93725 \]
Now substituting this value to find \(x\):
-
For the positive root: \[ x = 7.93725 - 16 \approx -8.06275 \quad \text{(rounding gives)} \quad -8.06 \]
-
For the negative root: \[ x = -7.93725 - 16 \approx -23.93725 \quad \text{(rounding gives)} \quad -23.94 \]
Thus, the two solutions to the equation \(\frac{(x + 16)^2}{9} = 7\) are:
\[ \boxed{-8.06} \quad \text{and} \quad \boxed{-23.94} \]