# The total profit function P(x) for a company producing x thousand units is given by P(x) = -3x^2 + 36x - 81. How do you find the values of x for which the company makes a profit?

Jul 24, 2015

I found: $3 <$$x$$< 9$ (in thousand units)
You can see that producing zero units ($x = 0$, i.e., the company is idle) you get a loss, $P \left(0\right) = - 81$ this is because you have to pay fixed costs, say, taxes, rent...etc.
There is a region where $P \left(x\right) > 0$ and you have a profit, where:
$- 3 {x}^{2} + 36 x - 81 > 0$ that gives you (using the Quadratic Formula):
$3 <$$x$$< 9$ (in thousand units) that described the red shaded region int the graph where $P \left(x\right) > 0$.