Question

Solve the equation with square roots, round to nearest hundredth
(x+16)^2/9 = 7

Answer 1

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

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

Next, we take the square root of both sides:

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

Calculating \(\sqrt{63}\):

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

We can also calculate \(3\sqrt{7}\) numerically. The approximate value of \(\sqrt{7}\) is about 2.6457513110645906, so:

\[
3\sqrt{7} \approx 3 \cdot 2.6457513110645906 \approx 7.937
\]

Now substituting back into our equation gives us:

\[
x + 16 = \pm 7.937
\]

This results in two cases:

1. \(x + 16 = 7.937\)
2. \(x + 16 = -7.937\)

**Case 1:**

\[
x + 16 = 7.937 \implies x = 7.937 - 16 \implies x \approx -8.063
\]

**Case 2:**

\[
x + 16 = -7.937 \implies x = -7.937 - 16 \implies x \approx -23.937
\]

Thus, we can summarize the solutions as:

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

Finally, rounding to the nearest hundredth, the solutions are:

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