Thread: Zero Product Property - Profits of a video game company.

I understand the technicality of doing zero product property (finding the two-intercepts and what not) but I just don't understand how it works or makes sense in real-life applications. As I'll illustrate with the question I'm having difficulty with. I have the two x-intercepts, I just don't know how to make sense of them in the context of the question.

9. The profit, P, of a video game company, in thousands of dollars, is given by P= -5x² + 550 x - 5000, where x is the amount spent on advertising in thousands of dollars.

a) Determine the amount spent on advertising that will result in a profit of $0. In other words, you will determine the amount that must be spent on advertising in order for the company to "break even."

My Solution:

P= -5x² + 550x - 5000
0= -5x² + 550x - 5000
= -5 (x² -110x + 1000)

x² is a product of = 1 x 1
1000 is a product of = -100, -10

1 x -100 = -100
1 x -10 = -10
= -110

equal to -110x

-5 (x-100)(x-10)

x-100 = 0 or x-10=0
x= 100 or x=10


My Issue

So I have the x-intercepts of 100 and 10 (as I can also see on the graph that they provided) but how are they answering the question??
So like.. does the company spend AT LEAST 10 (10, 000 dollars) to break even, and a maximum of 100 (100, 000 dollars) to break even?

And it still doesn't make sense. How can you have a profit of $0 and break even IF you're spending money in the first place. And how the hell do you know what your profits are if you're just SPENDING money!! Like you can't just accurately predict your profits.

So is the question really just asking me how much money they spent on advertising so they can know how much money they'll have to make IN sales to break even??
So do I solve the equation with both of the x-intercepts to get two different results. Uggh, can someone please just explain this to me? I'm getting a headache!

The solutions to your function is correct.

The profit P is equal to zero when x=100 or x=10. You might be thinking in terms of linearity (thinking the amount you spend on advertising is directly proportional to profit) but given your function P the function is parabolic, not linear. You have to consider market research and labor costs for example, which sometimes don't break even. You can also think in terms of supply and demand (the usual relationship between how much you have and how much people want is not necessarily linear).