Expert: Scott A Wilson - 12/17/2011

Question
Hi, I need major help with algebra. I'm doing a project in which I have to create a quadratic equation. I created a word problem:

In a place, for every 20 people who celebrate Xmas, there are 19 heartfuls of Xmas magic. As the population increases, for every 10 additional residents, about 0.1 heartfuls of Xmas magic are lost.
(10x+20)(-0.1x+19)
y=-x^2+188x+380
h=92, k=9216?

My question is, does this answer even make sense with the problem? Does this mean that when 92 people enter the place, the maximum amount of magic is 9216 heartfuls? I'm very confused and I need to get this dine by tonight. This is a big project. I still need to put everything on a decorative greeting card for the project. Please help me ASAP. Thank you so much.

Answer
Sorry the answer is so late and I'm not even sure this is on time, but that equation looks correct.  The maximum occurs when 2x = 188, so it is at x = 94.
If x = 94 is put in the equation, we get y = (-94+188)94 + 380.
Since -94 + 188 = 94, that is the same as 94^2 + 380.
Now 94^2 = 8836, so that is 8836 + 380 = 9216.

That number is recognized as 96^2, but I'm not sure that's relevant.
It is reconized since 9216 = 10,000 - 800 + 16, and that is the same as
100^2 - 2*4*100 + 4^2 = (100 - 4)^2 = 96^2.