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
Answers
GPT-4o mini
To determine how many solutions are in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), let's solve it step by step.
1. Start by simplifying the 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 \(x - 5 = 3\):
\[
x = 3 + 5 = 8
\]
From \(x - 5 = -3\):
\[
x = -3 + 5 = 2
\]
So, the solutions are \(x = 8\) and \(x = 2\).
6. Therefore, there are **two solutions** in the solution set.
The correct response is: **two**.
1. Start by simplifying the 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 \(x - 5 = 3\):
\[
x = 3 + 5 = 8
\]
From \(x - 5 = -3\):
\[
x = -3 + 5 = 2
\]
So, the solutions are \(x = 8\) and \(x = 2\).
6. Therefore, there are **two solutions** in the solution set.
The correct response is: **two**.