Solution Sets of Quadratic Equations Quick Check

Question

5 of 55 of 5 Items

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

infinitely many
infinitely many

one
one

zero
zero

two
two
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Answer 1

To determine the number of solutions for the equation \( 2(x-5)^2 + 2 = 20 \), we need to solve the equation step by step.

1. Start by isolating the quadratic term: \[ 2(x-5)^2 + 2 = 20 \]

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

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

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

4. Solve for \( x \) in both cases:

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

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

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

Since there are two distinct solutions, the correct response is:

two