Remark
Yesterday and today make it very dangerous to answer any math question. I'm going to see if I can answer this without the editor throwing up. The method is worked out now that I can see that the bug might have been fixed.
The final answer is y = -(x - 24)^2 + 144
The first thing to do is to complete the square.
y = -(x^2 - 48x ) - 432
y = -(x^2 - 48x + (48/2)^2 ) - 432 + 24^2
y = -(x - 24)^2 + 576 - 432
y = -(x - 24)^2 + 144
The vertex is (24,144)
Explanation
It means that when 24 games are sold, the profit is a maximum 144 dollars. You can convince yourself of that by trying a couple of numbers. Try x = 28. If you sell 28 games the following will happen.
y = -(28 - 24)^2 + 144
y = - 4^2 + 144
y = -16 + 144
y = 128 which is less that 144.
You could also try 20 which will give you the same answer 128
Problem B
What are the x intercepts?
y = -x^2 + 48x - 432
You do not need the quadratic equation for this. It factors.
y = -(x - 12)(x - 36)
when y = 0 then either
x - 12 = 0
x = 12
When x = 12 the profit is 0. He starts to make money at x = 13
or
x - 36 = 0
x = 36
also he his making nothing when x = 36 which seems odd because you would think that the more you sell the more you make, but the quadratic makes this a limiting consideration.
So x<12 loses money
x>36 looses money
12<x<36 is where he makes some money.
