Question
Solution Sets of Quadratic Equations Quick Check
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Question
Show how many solutions are in the solution set for 2(x−5)2+2=20.(1 point)
Responses
infinitely many
infinitely many
one
one
zero
zero
two
two
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Answers
GPT-4o mini
To determine the number of solutions for the equation \( 2(x-5)^2 + 2 = 20 \), we need to solve the equation step by step.
1. Start by isolating the quadratic term:
\[
2(x-5)^2 + 2 = 20
\]
Subtract 2 from both sides:
\[
2(x-5)^2 = 18
\]
2. Divide both sides by 2:
\[
(x-5)^2 = 9
\]
3. Now, take the square root of both sides:
\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\]
4. Solve for \( x \) in both cases:
- For \( x - 5 = 3 \):
\[
x = 3 + 5 = 8
\]
- For \( x - 5 = -3 \):
\[
x = -3 + 5 = 2
\]
Thus, the solutions are \( x = 8 \) and \( x = 2 \).
Since there are two distinct solutions, the correct response is:
**two**
1. Start by isolating the quadratic term:
\[
2(x-5)^2 + 2 = 20
\]
Subtract 2 from both sides:
\[
2(x-5)^2 = 18
\]
2. Divide both sides by 2:
\[
(x-5)^2 = 9
\]
3. Now, take the square root of both sides:
\[
x - 5 = 3 \quad \text{or} \quad x - 5 = -3
\]
4. Solve for \( x \) in both cases:
- For \( x - 5 = 3 \):
\[
x = 3 + 5 = 8
\]
- For \( x - 5 = -3 \):
\[
x = -3 + 5 = 2
\]
Thus, the solutions are \( x = 8 \) and \( x = 2 \).
Since there are two distinct solutions, the correct response is:
**two**