Question
the revenue function r(x) and the cost function c(x) for a particular product are given. these functions are valid only for the specified range of values.

find the number of units that must be produced to break even.
upper r(x) = 200 x -2 x squared

c(x) = −x squared + 25x+3100 0 less than or equal to x less than or equal to 100

Set R(x) equal to C(x).  Solve the resulting equation for x, paying attention to the domain of C(x):

Solve 200x-2x^2 = -x^2 + 25x + 31000 for x greater than or equal to zero but less than or equal to x.  That means you must reject any solution outside of this domain.

Rearrange all of these terms on the left side of your equation and place zero on the right side.  Next, rearrange all of the terms on the left side in descending order by powers of x.  Next, solve the resulting quadratic equation.  Make certain to reject any x that does not fall within [0,100].