Question

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

Answer 1

To solve the quadratic equation \(\frac{(x+16)^2}{9} = 7\), we will first eliminate the fraction by multiplying both sides by 9:

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

Next, we take the square root of both sides. Remember to consider both the positive and negative roots:

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

Calculating \(\sqrt{63}\):

\[
\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \approx 3 \cdot 2.64575 \approx 7.937
\]

Now we substitute this back into our equation:

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

Now we solve for \(x\) in each case.

1. For \(x + 16 = 3\sqrt{7}\):

\[
x = 3\sqrt{7} - 16 \approx 7.937 - 16 \approx -8.063
\]

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

\[
x = -3\sqrt{7} - 16 \approx -7.937 - 16 \approx -23.937
\]

Thus the solutions to the equation \(\frac{(x+16)^2}{9} = 7\) rounded to the nearest hundredth are:

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

So the final answers are:

\[
\boxed{-8.06} \quad \text{and} \quad \boxed{-23.94}
\]