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 \((x + 16)^2 / 9 = 7\), 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:

\[
\{-8.06, -23.94\}
\]