Question

What are the solutions to `\left(x+7\right)^{2}=81?`

Answer 1

What are the solutions to `2\left(x-7\right)^{2}=32?`

Answer 2

To solve the equation \((x + 7)^{2} = 81\), we will start by taking the square root of both sides.

\[
x + 7 = \pm 9
\]

This gives us two separate equations to solve:

1. \(x + 7 = 9\)
2. \(x + 7 = -9\)

**Solving the first equation:**

\[
x + 7 = 9
\]
Subtracting 7 from both sides:
\[
x = 9 - 7
\]
\[
x = 2
\]

**Solving the second equation:**

\[
x + 7 = -9
\]
Subtracting 7 from both sides:
\[
x = -9 - 7
\]
\[
x = -16
\]

Thus, the solutions to the equation \((x + 7)^{2} = 81\) are

\[
\boxed{2} \quad \text{and} \quad \boxed{-16}.
\]

Answer 3

To solve the equation \(2(x - 7)^{2} = 32\), we first divide both sides by 2:

\[
(x - 7)^{2} = \frac{32}{2}
\]

\[
(x - 7)^{2} = 16
\]

Next, we take the square root of both sides:

\[
x - 7 = \pm 4
\]

This gives us two separate equations to solve:

1. \(x - 7 = 4\)
2. \(x - 7 = -4\)

**Solving the first equation:**

\[
x - 7 = 4
\]
Adding 7 to both sides:
\[
x = 4 + 7
\]
\[
x = 11
\]

**Solving the second equation:**

\[
x - 7 = -4
\]
Adding 7 to both sides:
\[
x = -4 + 7
\]
\[
x = 3
\]

Thus, the solutions to the equation \(2(x - 7)^{2} = 32\) are

\[
\boxed{11} \quad \text{and} \quad \boxed{3}.
\]