Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x+16)^2/9=7

To solve the quadratic equation \((x + 16)^2 / 9 = 7\), follow these steps:

1. Multiply both sides by 9 to eliminate the fraction:

   \[ (x + 16)^2 = 7 \cdot 9 \]

   \[ (x + 16)^2 = 63 \]

2. Take the square root of both sides:

   \[ x + 16 = \pm \sqrt{63} \]

   We simplify \(\sqrt{63}\): \[ \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \]

   So, we have:

   \[ x + 16 = \pm 3\sqrt{7} \]

3. Solve for \(x\):

   \[ x = -16 \pm 3\sqrt{7} \]

4. Calculate the numerical values. First, find \(3\sqrt{7}\): \[ \sqrt{7} \approx 2.64575 \] Therefore, \[ 3\sqrt{7} \approx 3 \times 2.64575 \approx 7.93725 \]

   Now calculate the two possible values for \(x\):

   \[ x_1 = -16 + 3\sqrt{7} \approx -16 + 7.93725 \approx -8.06275 \approx -8.06 \]

   \[ x_2 = -16 - 3\sqrt{7} \approx -16 - 7.93725 \approx -23.93725 \approx -23.94 \]

Thus, the solutions to the equation are:

\[ x \approx -8.06 \quad \text{and} \quad x \approx -23.94 \]

So, filling in the answers:

\[ {-8.06, -23.94} \]