Question

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

two
two

zero
zero

infinitely many
infinitely many

one

Answer 1

To solve the equation \( 2(x-5)^2 + 2 = 20 \), we first simplify it step by step.

1. Subtract 2 from both sides:
\[
2(x-5)^2 = 20 - 2
\]
\[
2(x-5)^2 = 18
\]

2. Divide both sides by 2:
\[
(x-5)^2 = \frac{18}{2}
\]
\[
(x-5)^2 = 9
\]

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

4. Solving for \( x \):
- For the positive root:
\[
x - 5 = 3 \implies x = 3 + 5 = 8
\]
- For the negative root:
\[
x - 5 = -3 \implies x = -3 + 5 = 2
\]

So, the solutions to the equation are \( x = 8 \) and \( x = 2 \).

Thus, there are **two** solutions in the solution set.

The correct response is:
**two**