So the question is:

A theatre with $1000$ seats can be filled if tickets are $12$ but for every increase in $25$ cents, the theatre will lose $5$ attendees. What is the cost of the ticket that leads to the maximum profit?

I've found this solution and I'm left with the equation below:

Let $x$ be the number of price increases

$y$ is the total profits

$y = -1.25(x-76)^2 + 19220$

Now my question is, why is $76$ the number of price increases? Is there a special rule that makes $(x-76) 0$?

**EDIT:**

$y = (12+0.25x)(1000-5x)$

$y = 12000-60x+250x-1.25x^2$

$y = -1.25x^2+190x+12000$

$y = -1.25x(x^2-152x)$

$y = -1.25(x^2-152x+5776-5776)+12000$

$y = -1.25(x^2-152x+5776)+7220+12000$

$y = -1.25(x-76)^2+19220$