Question

How many solutions are in the solution set for 2(x-5)^2+2=20
one
infinitely many
two
zero

Answer 1

To solve the equation \( 2(x - 5)^2 + 2 = 20 \), we first simplify it step by step.

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 \(x\)**:
- From \(x - 5 = 3\):
\[
x = 3 + 5 = 8
\]
- From \(x - 5 = -3\):
\[
x = -3 + 5 = 2
\]

Thus, the solutions are \(x = 8\) and \(x = 2\).

Therefore, there are **two solutions** in the solution set for the given equation.

The answer is **two**.