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 start by eliminating the fraction. We can do this by multiplying both sides of the equation by 9:

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

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

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

Calculating \(\sqrt{63}\):

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

Now substituting back, we have:

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

Next, we isolate \(x\):

1. For the positive case:
\[
x = 3\sqrt{7} - 16
\]

2. For the negative case:
\[
x = -3\sqrt{7} - 16
\]

Now we will compute the numerical values:

First, we need the approximate value of \(\sqrt{7}\):

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

Calculating \(3\sqrt{7}\):

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

Now substituting this value to find \(x\):

1. For the positive root:
\[
x = 7.93725 - 16 \approx -8.06275 \quad \text{(rounding gives)} \quad -8.06
\]

2. For the negative root:
\[
x = -7.93725 - 16 \approx -23.93725 \quad \text{(rounding gives)} \quad -23.94
\]

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

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