Question

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

one
one

zero
zero

infinitely many
infinitely many

two

Answer 1

To find the number of solutions in the solution set for the equation \(2(x - 5)^2 + 2 = 20\), we first need to simplify and solve the equation.

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

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

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

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

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

The correct response is: **two**.