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

Question

2
(
𝑥
−
5
)
2
+
2
=
20
.(1 point)
Responses

infinitely many
infinitely many

two
two

one
one

zero

Answer 1

To solve the equation \( 2(x - 5)^2 + 2 = 20 \), we can start by isolating the quadratic term.

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

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

3. Take the square root of both sides: \[ x - 5 = 3 \quad \text{or} \quad x - 5 = -3 \]

4. Solving these gives: \[ x = 8 \quad \text{or} \quad x = 2 \]

Thus, there are two solutions \( (x = 8 \text{ and } x = 2) \) in the solution set.

Therefore, the correct response is two.