Question

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

two
two

one
one

infinitely many
infinitely many

zero

Answer 1

To determine how many solutions are in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), we can first simplify and solve the equation step by step.

1. Start by isolating the term with the square:
\[
2(x - 5)^2 + 2 = 20
\]
Subtract 2 from both sides:
\[
2(x - 5)^2 = 18
\]
Now divide both sides by 2:
\[
(x - 5)^2 = 9
\]

2. Next, take the square root of both sides:
\[
x - 5 = \pm 3
\]

3. This gives us two separate equations to solve:
\[
x - 5 = 3 \quad \text{and} \quad x - 5 = -3
\]

4. Solving these equations:
- For \( x - 5 = 3 \):
\[
x = 8
\]
- For \( x - 5 = -3 \):
\[
x = 2
\]

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

Therefore, there are **two solutions** in the solution set for the original equation.

The correct response is:
**two**