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To determine how many solutions are in the solution set for the equation \(2(x - 5)^2 + 2 = 20\), we can start by simplifying it step-by-step.
1. **Subtract 2 from both sides of the equation**:
\[
2(x - 5)^2 + 2 - 2 = 20 - 2
\]
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**:
\[
x - 5 = \pm 3
\]
This leads to two equations:
- \(x - 5 = 3\)
- \(x - 5 = -3\)
4. **Solve each equation**:
From \(x - 5 = 3\):
\[
x = 3 + 5 = 8
\]
From \(x - 5 = -3\):
\[
x = -3 + 5 = 2
\]
Thus, the solutions are \(x = 8\) and \(x = 2\).
Since we found two distinct solutions, the number of solutions in the solution set is **two**.