To find the number of solutions in the solution set for the equation \( 2(x - 5)^2 + 2 = 20 \), we can first simplify the equation and solve for \( x \).
Starting with:
\[ 2(x - 5)^2 + 2 = 20 \]
Subtract 2 from both sides to get:
\[ 2(x - 5)^2 = 18 \]
Now, divide by 2:
\[ (x - 5)^2 = 9 \]
Next, take the square root of both sides. Remember that when you take the square root of both sides, you get two solutions because both the positive and negative square roots are solutions to the squared variable.
\[ x - 5 = \pm 3 \]
This gives us two equations to solve for \( x \):
1. \( x - 5 = 3 \), which simplifies to \( x = 8 \)
2. \( x - 5 = -3 \), which simplifies to \( x = 2 \)
Therefore, the equation \( 2(x - 5)^2 + 2 = 20 \) has two distinct solutions in its solution set.
The correct response is:
two
Show how many solutions are in the solution set for 2(x−5)2+2=20 .(1 point)
Responses
zero
zero
infinitely many
infinitely many
two
two
one
one
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