Question

Show how many solutions are in the solution set for the equation 2(x - 5)^2 + 2 = 20.
A) infinitely many
B) one
C) zero
D) two

Answer 1

To find how many solutions are in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), we can solve it step by step:

1. Start with the original equation:
\[
2(x - 5)^2 + 2 = 20
\]

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

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

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

5. Solve for \( x \):
- From \( x - 5 = 3 \):
\[
x = 8
\]
- From \( x - 5 = -3 \):
\[
x = 2
\]

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

Since there are two distinct solutions, the answer is:
\[
\text{D) two}
\]