Question

Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x+16)2/9 =7 .(1 point)

Answer 1

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

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

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

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

2. **Take the square root of both sides**:

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

\(\sqrt{63}\) can be simplified:

\[
\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}
\]

3. **Now, we can express the equations:**

\[
x + 16 = 3\sqrt{7} \quad \text{or} \quad x + 16 = -3\sqrt{7}
\]

4. **Solve for \(x\)** in both cases:

- First case:

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

- Second case:

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

5. **Calculating the values**:
- First, calculate \(3\sqrt{7}\):

\[
3\sqrt{7} \approx 3 \times 2.64575 \approx 7.93725
\]

- For \(x = 3\sqrt{7} - 16\):

\[
x \approx 7.93725 - 16 \approx -8.06275
\]
(Rounding to the nearest hundredth: \(-8.06\))

- For \(x = -3\sqrt{7} - 16\):

\[
-3\sqrt{7} \approx -7.93725
\]
So,

\[
x \approx -7.93725 - 16 \approx -23.93725
\]
(Rounding to the nearest hundredth: \(-23.94\))

6. **Final Answers**:

The solutions to the equation are:
\[
x \approx -8.06 \quad \text{and} \quad x \approx -23.94
\]