Fan and Medal!
The following function represents the profit P(n), in dollars, that a concert promoter makes by selling tickets for n dollars each:
P(n) = -250n^2 + 2,500n - 4,000
Part A: What are the zeroes of the above function, and what do they represent? Show your work. (4 points)
Part B: Find the maximum profit by completing the square of the function P(n). Show the steps of your work. (4 points)
Part C: What is the axis of symmetry of the function P(n)? (2 points)

- anonymous

- katieb

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- anonymous

\[P(n) = -250n^2 + 2,500n - 4,000 \] you want the zeros?

- anonymous

I already finished part A
I know the 0's are 2 and 8

- anonymous

no that is wrong

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## More answers

- anonymous

\(n\) is the price
i guess if he prices the tickets at $2 or at $8 he makes no money

- anonymous

The zeroes of the function would be 2 and 8. The zeroes of the function above represents the specific prices, 2 and 8, which will result in a zero profit.

- anonymous

is that correct?

- anonymous

yes looks good to me

- anonymous

Okay (:

- anonymous

Now how do I "complete the square"

- anonymous

there is a much easier way to do it (find the max) than completing the square, but we can do it that way too if you have to

- anonymous

first of all it is pretty obvious that if it is zero at 2 and at 8 then it is the biggest half way between at 5

- anonymous

plus the first coordinate of the vertex is always
\[-\frac{b}{2a}\] which in your case is
\[-\frac{2500}{2\times (-250)}=5\]

- anonymous

Yeah, but it specifically asks to complete the square (which I hate because it's so confusing)

- anonymous

but we can still complete the square if you are dying to do it

- anonymous

factor out the \(-250\) from the first two terms

- anonymous

\[ -250n^2 + 2,500n - 4,000 \]
\[-250(n^2-10n)-4,000\]

- anonymous

P(n)=-250(n^2-10n)-4000?

- anonymous

Okay i got that aha

- anonymous

right that is step one

- anonymous

then half of \(-10\) is \(-5\) complete the square via
\[-250(n-5)^2-4000+25\times 250\]

- anonymous

Yeah... I got lost there ):

- anonymous

yeah i can see why

- anonymous

when you replace \(n^2-10n\) by \(n-5)^2\) you have changed it

- anonymous

\[(n-5)^2=n^2-10n+25\] so you have added \(25\) when you make the change

- anonymous

Ohh, okay I see that now, but like if it's so many numbers and such, my brain likes to just stop reading it

- anonymous

why 25 though?

- anonymous

because when you square \(n-5\) you get \((n-5)^2=n^2-10n+25\)

- anonymous

Ohhh

- anonymous

Okay (:

- anonymous

now don't forget the \(-250\) outside the parentheses

- anonymous

so actually you have subtracted \(25\times 250\) so you have to add it back

- anonymous

hence \[-250(n-5)^2-4000+25\times 250\]

- anonymous

but why the times symbol?

- anonymous

what else would you like me to write? look at this line \[-250(n-5)^2\]

- anonymous

if you multiply that out you get
\[-250(n^2-10n+25)\] as a first step

- anonymous

I meant the times symbol for 25 x 250

- anonymous

and when you multiply that out, you get \[-250n^2+2500n-6250\]

- anonymous

you subtracted \(6250\) from the original expression, so you have to add it back

- anonymous

there is really a much much easier way to do it

- anonymous

Yeah, but my algebra teachers just love making us suffer haha

- anonymous

since you know that the first coordinate of the vertex is \(5\) if you want the second coordinate, all you have to do is find \(P(5)\)

- anonymous

you can pretend to to it the other way, but then just find \(P(5)\) and write that out at the end

- anonymous

so after P(n)=-250(n-5)^2-4000+25*250 ,what do I do, or is that the final step?

- anonymous

algebra teachers lead sad lives, most are either frustrated mathematicians or are teaching at the outer edge of their knowledge

- anonymous

compute the number at the end

- anonymous

-250(n-5)^2-4000+6250

- anonymous

\[-400+25\times 250=2250\]

- anonymous

-250(n-5)^2+2250

- anonymous

yes

- anonymous

And now? do I make it (n-5)(n-5)?

- anonymous

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