Question

Show how many solutions are in the solution set for 2(x−5)^2 + 2=20
.(1 point)
Responses

zero
zero

one
one

infinitely many
infinitely many

two

Answer 1

To determine how many solutions are in the solution set for the equation \(2(x-5)^2 + 2 = 20\), we can follow these steps:

1. **Subtract 2 from both sides**:
\[
2(x-5)^2 + 2 - 2 = 20 - 2
\]
This simplifies to:
\[
2(x-5)^2 = 18
\]

2. **Divide both sides by 2**:
\[
(x-5)^2 = 9
\]

3. **Take the square root of both sides**:
\[
x-5 = 3 \quad \text{or} \quad x-5 = -3
\]

4. **Solve for \(x\)**:
- For \(x - 5 = 3\):
\[
x = 3 + 5 = 8
\]
- For \(x - 5 = -3\):
\[
x = -3 + 5 = 2
\]

Thus, we have two solutions: \(x = 8\) and \(x = 2\).

Therefore, the answer is:
**two**.