Question
Show how many solutions are in the solution set for 2(x−5)2+2=20 .(1 point)
Responses
zero
zero
two
two
infinitely many
infinitely many
one
one
Responses
zero
zero
two
two
infinitely many
infinitely many
one
one
Answers
GPT-4o mini
To find how many solutions are in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), we can solve the equation step by step.
1. **Isolate the squared term:**
\[
2(x - 5)^2 + 2 = 20
\]
Subtract 2 from both sides:
\[
2(x - 5)^2 = 18
\]
2. **Divide by 2:**
\[
(x - 5)^2 = 9
\]
3. **Take the square root of both sides:**
\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\]
4. **Solve for \(x\):**
- From \(x - 5 = 3\):
\[
x = 8
\]
- From \(x - 5 = -3\):
\[
x = 2
\]
Thus, the equation has two solutions: \(x = 8\) and \(x = 2\).
The correct response is **two**.
1. **Isolate the squared term:**
\[
2(x - 5)^2 + 2 = 20
\]
Subtract 2 from both sides:
\[
2(x - 5)^2 = 18
\]
2. **Divide by 2:**
\[
(x - 5)^2 = 9
\]
3. **Take the square root of both sides:**
\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\]
4. **Solve for \(x\):**
- From \(x - 5 = 3\):
\[
x = 8
\]
- From \(x - 5 = -3\):
\[
x = 2
\]
Thus, the equation has two solutions: \(x = 8\) and \(x = 2\).
The correct response is **two**.