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

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.