Question

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

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Answer 1

[AI-generated answer — Bot GPT-4 Turbo]

Let's solve the equation step by step to find the number of solutions in the solution set for the given equation:

2(x−5)² + 2 = 20

Start by isolating the quadratic term:

2(x−5)² = 20 − 2
2(x−5)² = 18

Now, divide both sides by 2 to get the (x−5)² term alone:

(x−5)² = 18 / 2
(x−5)² = 9

Now we will take the square root of both sides to solve for x. Remember that when you take the square root of both sides of an equation, you get two solutions, one positive and one negative:

√(x−5)² = ±√9
x−5 = ±3

Now solve for x by adding 5 to both sides of the equation for each solution:

Positive root:
x−5 = 3
x = 3 + 5
x = 8

Negative root:
x−5 = -3
x = -3 + 5
x = 2

The equation has two solutions, x = 8 and x = 2, so the solution set has 2 solutions.