To find the zeros of the function [tex]\( y = -2(x-2)^2 + 6 \)[/tex] which represents the daily profit of a hot dog stand in terms of the price [tex]\( x \)[/tex] of a hot dog, we need to solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].
### Step-by-Step Solution
1. Set the function equal to zero:
[tex]\[
-2(x-2)^2 + 6 = 0
\][/tex]
2. Isolate the quadratic term:
[tex]\[
-2(x-2)^2 = -6
\][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
(x-2)^2 = 3
\][/tex]
3. Solve for [tex]\( x \)[/tex] by taking the square root of both sides:
[tex]\[
x - 2 = \pm \sqrt{3}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 2 \pm \sqrt{3}
\][/tex]
So, the zeros of the function are [tex]\( x = 2 - \sqrt{3} \)[/tex] and [tex]\( x = 2 + \sqrt{3} \)[/tex].
The interpretation of these zeros is that they represent the hot dog prices at which the profit is exactly \[tex]$0.00 (no profit).
### Answers
- For the zeros: \( x = 2 \pm \sqrt{3} \).
Correct answer is A.
- For the interpretation: The zeros are the hot dog prices that give $[/tex]\[tex]$ 0.00$[/tex] profit (no profit).
Correct answer is D.
