Show how many solutions are in the solution set for 2(x−5)2+2=20

Question

.(1 point)
Responses

zero
zero

two
two

one
one

infinitely many
infinitely many

Answer 1

To solve the equation \( 2(x - 5)^2 + 2 = 20 \), we first simplify it step by step.

1. Subtract 2 from both sides: \[ 2(x - 5)^2 = 20 - 2 \] \[ 2(x - 5)^2 = 18 \]

2. Divide both sides by 2: \[ (x - 5)^2 = \frac{18}{2} \] \[ (x - 5)^2 = 9 \]

3. Now, take the square root of both sides: \[ x - 5 = \pm 3 \]

This gives us two equations to solve:

1. \( x - 5 = 3 \) \[ x = 3 + 5 = 8 \]

2. \( x - 5 = -3 \) \[ x = -3 + 5 = 2 \]

Thus, the two solutions are \( x = 8 \) and \( x = 2 \).

In conclusion, the solution set has two solutions.

So the response is: two