Show 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 can start by simplifying it step by step.

1. Subtract 2 from both sides: \[ 2(x - 5)^2 = 18 \]

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

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

Therefore, the answer is two.