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 \((x + 16)^2 / 9 = 7\), we first want to eliminate the fraction by multiplying both sides by 9:

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

Calculating the right side:

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

Next, we take the square root of both sides:

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

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

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

So we have:

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

Now we can solve for \(x\) by isolating it:

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

Now we'll compute the two possible solutions:

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

First, we need to calculate \(3\sqrt{7}\):

\(\sqrt{7} \approx 2.64575\)

Thus:

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

Now subtract and add this value to \(-16\):

1. \(x_1 = -16 + 7.93725 \approx -8.06275\)
2. \(x_2 = -16 - 7.93725 \approx -23.93725\)

We will round the results to the nearest hundredth:

1. \(x_1 \approx -8.06\)
2. \(x_2 \approx -23.94\)

Thus, the solutions to the quadratic equation are:

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