Question

May I please have some help? A medal and a fan will be given. :)

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.

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

\[\Large\rm P(n)=-250n^2 + 2500n - 4000\]So the find `zeros`, we set the P(n) equal to zero.\[\Large\rm 0=-250n^2 + 2500n - 4000\]This will be easier to work with if we have no fancy coefficient on the squared term.
So let's factor a -250 out of each term:\[\Large\rm 0=-250(n^2-10n+16)\]Then divide the -250 away,\[\Large\rm 0=n^2-10n+16\]It looks like this will factor nicely!
Confused by any of those steps?

Answer 2

Hold on one second, let me look really quickly. :)

Answer 3

Nope! I understand for sure! So, I find the GCF for the 16?

Answer 4

Factors of 16 that add to -10.

So we want two numbers that multiply to give us positive,
but add to give us negative,
So both factors should be negative.

Can you find the right combination? :o

Answer 5

Yes, I can hehehehehe...give me 3 seconds.

Answer 6

\[-8 - 2 = - 10/ -8 • -2 = +16\]

Answer 7

So, \[0 = n^2 - 8n - 2n + 16\]

Answer 8

I'm alone again...I can't wait to get on the road again! :D

Answer 9

Mmmm ok if you're going to break it up like that, I guess we can factor by grouping.

Answer 10

I found my answer! Would you like to see the process?

Answer 11

\[n^2 - 8n\]
\[n^2, n\]
\[n ( n - 8 )\]

Answer 12

\[\Large\rm 0 = \color{royalblue}{n^2 - 8n} \color{orangered}{- 2n + 16}\]\[\Large\rm 0 = \color{royalblue}{n(n - 8)} \color{orangered}{- 2n + 16}\]k looks good so far,

Answer 13

\[-2n + 16\]
\[-2: 2 • 1\]
\[16: 8 • 2\]
\[GCF: -2 ( n - 8 )\]

Answer 14

\[( n - 2 ) ( n - 8 )\]

Answer 15

Cool. so our zeros are what?

n=?

Answer 16

Our zeroes, good sir/ma'am, are +2, and +8.

Answer 17

Mmm k cool :)
What do you think these values might represent?

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

Answer 18

I think that they represent the amount of dollars the promoter earns from a concert per ticket sold?

Answer 19

That's the general representation of the function and n.
That' doesn't answer part A though.

So we set our P(n) equal to zero.
So these are the specific ticket prices, `n`, which result in ZERO profit.
Understand?

Answer 20

Oh, so they represent the specific prices, +2 and +8, which will result in a zero profit? Is that my answer with the work as well?

Answer 21

Yes, part A is that reponse + the work we did.

2$ wasn't a high enough price. We didn't charge enough, didn't make any profit.
8$ per ticket was too expensive, not enough people bought tickets or something.

Answer 22

Ooo we've gotta complete the square >.<

Answer 23

Ah yes Part B:....how I am going to not miss doing this. XD

Answer 24

Going back to our original expression:\[\Large\rm P(n)=\color{royalblue}{-250n^2 + 2500n} - 4000\]We want to complete the square on this blue part.

Answer 25

Ok

Answer 26

Again, factoring out a -250 gives us:\[\Large\rm P(n)=\color{royalblue}{-250(n^2 - 10n)} - 4000\]

Answer 27

The part inside the brackets, do you remember how to complete the square?

Answer 28

I think I'm supposed to distribute it, right?

Answer 29

You take half of the b term, and square it.
That's the value that completes the square.

So we have -10 and we need to cut it half and then square it.

Answer 30

OOOOohhhhhh.....I was doing this earlier, and I never understood what to do next.

Answer 31

I remember this part, would you mind helping me with this? It's a little difficult with these numbers, hehe. ^^'

Answer 32

So half of -10 is -5.
Then squaring it tells us that positive 25 is the value we want to complete the square, yes?

Answer 33

\[P (n) = −250 ( n^2 − 10n + 25 ) −4,000\]

Answer 34

Yes

Answer 35

\[\Large\rm P(n)=-250(n^2 - 10n\color{orangered}{+25}) - 4000\]Ok good.
But do you see the mistake we made?
In the land of math, we can't just add 25 willy nilly.
We have to keep things balanced, yes?

Answer 36

Correct, we need to subtract 25 from both sides.

Answer 37

4,000 - 25 = 3,975

Answer 38

No no this is where we need to be very careful!
We didn't add 25 to the right side.
We added 25 inside the brackets yes? ( which results in a different number when taken out of the brackets)

So we need to subtract 25 in the brackets as well.

Answer 39

\[\Large\rm P(n)=-250(n^2 - 10n+25-25) - 4000\]

Answer 40

Wait...then we do what I said above right? Then-wait. I got it. -4000 - 25. I forgot the negative sign on the 4,000. HA Algebra is a killer! XD

Answer 41

Our goal is to end up with only this blue part in the brackets:\[\Large\rm P(n)=-250(\color{royalblue}{n^2 - 10n+25}-25) - 4000\]Then we'll be able to write it as a perfect square.
So we need to pull the -25 out of the brackets somehow.

Answer 42

Ok continue...

Answer 43

No when we pull the -25 out of the brackets, we have to multiply it by the -250 right?

Answer 44

So it turns into a very different number.

Answer 45

Right

Answer 46

Yeah, like over 5,000 I believe.

Answer 47

\[\Large\rm P(n)=-250(\color{royalblue}{n^2 - 10n+25})+6250 - 4000\]Yah, not quite over 9000 ;) but yes.

Answer 48

Haha! I got the 9,000 part! XD I understand what to do!

Answer 49

So, the answer would be:

Answer 50

\[P( n ) = -250 ( n^2 - 10n + 25 ) 2,250\]

Answer 51

\[P( n ) = -250 ( n^2 - 10n + 25 ) +2,250\]Ok good. Now we need to write the bracketed portion as a perfect square.

Answer 52

Could you show me?

Answer 53

@zepdrix

Answer 54

Well we could factor by grouping again if you're more comfortable with that.

Answer 55

Ok, let's do that!

Answer 56

\[\Large\rm n^2-10n+25\]

Answer 57

Do it!! >.<

Answer 58

brb i need some chocolate milk :U

Answer 59

Ok...it's good for you heart!

Answer 60

\[n^2 - 5n - 5n + 25\]
\[n ( n - 5 )\]
\[-5 ( n - 5 )\]
\[( n - 5 )^2\]

Answer 61

Bum bum!

Answer 62

\[\Large\rm P( n ) = -250 (n-5)^2 +2,250\]Cool.

Answer 63

What shape does this function make?

Answer 64

Hold on, let me think...

Answer 65

A parabola?

Answer 66

Mmm ok good, and since the leading term is negative, it's opening downward right?