Question

how many solutions are in the solution set for 2(x−5)2+2=20 .

Answer 1

To solve the equation \( 2(x - 5)^2 + 2 = 20 \), we start by isolating the squared term.

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

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

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

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

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

In conclusion, there are **2 solutions** in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \).