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

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

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

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

So, we have:

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

Now, we will solve for \(x\) in both cases:

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

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

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

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

Next, we will calculate the numerical values for both solutions. First, we find \(\sqrt{7}\):

\(\sqrt{7} \approx 2.6457513110645906\)

Now, substituting that into the solutions:

1. Calculating \(x = 3\sqrt{7} - 16\):

\[
x \approx 3(2.6457513110645906) - 16
\]
\[
x \approx 7.9372539331937718 - 16
\]
\[
x \approx -8.062746066806229
\]

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

\[
x \approx -3(2.6457513110645906) - 16
\]
\[
x \approx -7.9372539331937718 - 16
\]
\[
x \approx -23.93725393319377
\]

Thus, the two solutions rounded to the nearest hundredth are:

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

Final answers:

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