Show how many solutions are in the solution set for 2(x−5)2+2=20.(1 point)

Question

Responses

one
one

zero
zero

infinitely many
infinitely many

two

Answer 1

To find the number of solutions for the equation \( 2(x-5)^2 + 2 = 20 \), we can follow these steps:

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

4. Solve for \( x \):

   • For \( x-5 = 3 \): \[ x = 3 + 5 = 8 \]

   • For \( x-5 = -3 \): \[ x = -3 + 5 = 2 \]

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

Conclusion: There are two solutions in the solution set.

The answer is two.