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 $(x-76) 0$?
$y = (12+0.25x)(1000-5x)$
$y = 12000-60x+250x-1.25x^2$
$y = -1.25x^2+190x+12000$
$y = -1.25x(x^2-152x)$
$y = -1.25(x^2-152x+5776-5776)+12000$
$y = -1.25(x^2-152x+5776)+7220+12000$
$y = -1.25(x-76)^2+19220$