Question
Show how many solutions are in the solution set for 2 * (x - 5) ^ 2 + 2 = 20 (1 point) zero Infinitely many two one
Answers
GPT-4o mini
To find the number of solutions to the equation \( 2 \cdot (x - 5)^2 + 2 = 20 \), we'll first simplify and solve the equation step by step.
1. Subtract 2 from both sides:
\[
2 \cdot (x - 5)^2 + 2 - 2 = 20 - 2
\]
This simplifies to:
\[
2 \cdot (x - 5)^2 = 18
\]
2. Next, 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 \( x \):
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 final answer is:
**Two**.
1. Subtract 2 from both sides:
\[
2 \cdot (x - 5)^2 + 2 - 2 = 20 - 2
\]
This simplifies to:
\[
2 \cdot (x - 5)^2 = 18
\]
2. Next, 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 \( x \):
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 final answer is:
**Two**.