Question

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

Answer 1

To solve the equation \(\frac{(x+16)^2}{9} = 7\), we start by eliminating the fraction. We can do this by multiplying both sides by 9:

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

Calculating the right side gives:

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

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

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

Next, we can simplify \(\sqrt{63}\):

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

Now we have two equations to solve:

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

For the first equation:

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

Subtracting 16 from both sides gives:

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

For the second equation:

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

Subtracting 16 from both sides gives:

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

Next, we will approximate the values by calculating \(\sqrt{7}\):

\[
\sqrt{7} \approx 2.64575
\]

Now we can find \(3\sqrt{7}\):

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

Substituting this into the first equation:

\[
x \approx 7.93725 - 16 \approx -8.06275
\]

For the second equation:

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

So,

\[
x \approx -7.93725 - 16 \approx -23.93725
\]

Rounding to the nearest hundredth, we have:

1. \(x \approx -8.06\)
2. \(x \approx -23.94\)

Thus, the solutions to the equation are:

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