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

Answer 1

To find the number of solutions for the equation \( 2(x - 5)^2 + 2 = 20 \), we first simplify it.

1. Start by subtracting 2 from both sides:

\[ 2(x - 5)^2 = 20 - 2 \] \[ 2(x - 5)^2 = 18 \]

2. Next, divide both sides by 2:

\[ (x - 5)^2 = \frac{18}{2} \] \[ (x - 5)^2 = 9 \]

3. Take the square root of both sides:

\[ x - 5 = \pm 3 \]

This gives us two equations to solve:

• \( x - 5 = 3 \)
• \( x - 5 = -3 \)

4. Solving these equations:

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

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

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

Therefore, the number of solutions in the solution set is two.