Question

The following function represents the profit P(n), in dollars, that a concert promoter makes by selling tickets for n dollars each:
P(n) = -250n2 + 2,500n - 5,250

Part A: What are the zeroes of the above function, and what do they represent? Show your work.

Part B: Find the maximum profit by completing the square of the function P(n). Show the steps of your work.

Part C: What is the axis of symmetry of the function P(n)?

Answer 1

@Mehek14

Answer 2

\[P(n) = -250n^2 + 2,500^n - 5,250 \] factor out the \(250\) first, that will make finding the zeros easier

Answer 3

actually even better is to factor out the \(-250\)

Answer 4

@satellite73

Answer 5

would it be -250(n^2 + 10 - 21)?

Answer 6

@mathmale

Answer 7

Is the correct answer -250(n^2 - 10 +21)?

Answer 8

"Part A: What are the zeroes of the above function, and what do they represent? Show your work." This involves setting the given profit function = to 0 and solving for the zeros / roots / solutions.

First, however, I'd like for you to verify that you've copied down this problem correctly:\[P(n) = -250n2 + 2,500n - 5,250\]

Answer 9

ugh sooorrry I alays forget to put the ^ sumbol its supposed to be -250n^2 + 2500n - 5250

Answer 10

should be written as P(n) = -250n^2 + 2,500n - 5,250. This is a quadratic that's pretty straightforward to solve for roots. Somehow your P(n) = -250n2 + 2,500n - 5,250
became \[P(n) = -250n^2 + 2,500^n - 5,250,\]

Answer 11

which is dramatically different from a quadratic because you're using n as an exponent.

Try again. What is the equation after you've factored out the 250?

Answer 12

would it be
-250(n^2 - 10 + 21)
-250(n -7)(n-3)

Answer 13

You need to find the roots / zeros / solutions of

P(n) = -250n2 + 2,500n - 5,250 = 250(-n^2 + 10n - 21) = 0.

Answer 14

Be certain you know why your 10 is not correct, but 10n would be.

Answer 15

right sorry. So do I factor completely or do I factor the 250 and then make the new equation equal 0?

Answer 16

Yes. Factor out -250, actually. Once you've done that, set the resulting expression = to zero and factor it (or use some other method to solve this quadratic equation.

P(n)=-250( ???? ) = 0

Answer 17

it would be p(n) = -250(n^2 - 10n + 21)

Answer 18

and = 0 would come after

Answer 19

but how do I factor that?

Answer 20

Focus on your n^2 - 10n + 21 = 0.

Look at the last term; it's 21. What are possible factors, either + or -, of 21? Hopefully your factors, when added together, will sum up to -10n.

Answer 21

the two factors would be -7 and -3

Answer 22

Yes. Multiply them together, and you get +21 (which is the 3rd term of your quadratic).
combine them and you get -10n, which is the 2nd term of your quadratic. So far, so good!

Answer 23

Actually, "the two factors would be -7 and -3" is not quite right; you meant:

"The two roots would be -7 and -3.

If -7 is a root, a factor of that quadratic would be (x+7).

Answer 24

then would the new equation be (n - 3)(n - 7)?

Answer 25

yes

Answer 26

or rather \[(n-3)(n-7)=0\] which you can solve in your head for \(n\)

Answer 27

Fairy: Earlier, you typed "the two factors would be -7 and -3" I should have caught that. The roots are 7 and 3, and the factors are (x-7) and (x-3). To verify this, multiply those two factors together.

Answer 28

How would you now find the axis of symmetry of this parabolic graph?

Hint: have you seen this formula before? x=-b/(2a)

Answer 29

Yes but is this still for part a?

Answer 30

would the answer to part a be the zeroes are -7 and -3 or 7 and 3?

Answer 31

@mathmale

Answer 32

I think it would be 7 and 3 right since money can't be negative?

Answer 33

yes good point

Answer 34

so 7 and 3 are the two zeroes?

Answer 35

yes but be careful

Answer 36

Part A: What are the zeroes of the above function, and what do they represent? Show your work.

Answer 37

they make no money in profit if they sell the tickets for 3 or 7 dollars

Answer 38

what do you mean?

Answer 39

you make a good point but part a is asking for the zeroes of the function

Answer 40

The following function represents the profit P(n), in dollars, that a concert promoter makes by selling tickets for n dollars each:

Answer 41

if they sell for \(3\) then \(P(3)=0\) they make no profit
if they sell for \(7\) then \(P(7)=0\) still no profit

Answer 42

oh ok thanks! can you help me with part b?

Answer 43

sure

Answer 44

first off the answer is obvious
the function is a quadratic that opens down
it is zero at 3 and 7, so it has a maximum half way between them
what is half way between 3 and 7?