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 \(\frac{(x+16)^2}{9} = 7\), we will first isolate the squared term by multiplying both sides by 9:

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

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

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

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

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

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

So we have:

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

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

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

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

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

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

Next, we will calculate the approximate values of \(3\sqrt{7}\) and \(-3\sqrt{7}\) to find the values of \(x\).

First, let's calculate \(3\sqrt{7}\):

\[
\sqrt{7} \approx 2.6457513110645906 \quad \Rightarrow \quad 3\sqrt{7} \approx 3 \times 2.6457513110645906 \approx 7.9372539331
\]

So,

\[
x \approx 7.9372539331 - 16 \approx -8.0627460669 \quad \Rightarrow \quad x \approx -8.06 \quad (\text{rounded to the nearest hundredth})
\]

Now for the second case:

\[
-3\sqrt{7} \approx -7.9372539331 \quad \Rightarrow \quad x \approx -7.9372539331 - 16 \approx -23.9372539331 \quad \Rightarrow \quad x \approx -23.94 \quad (\text{rounded to the nearest hundredth})
\]

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

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