To break even the cost to produce the product, #c(x)#, must equal the revenue generated by selling the product, #r(x)#. Therefore,
#c(x) = r(x)#
or
#5x + 18 = 2x^2#
We can now solve this for #x#.
#5x + 18 - color(red)(5x) - color(blue)(18) = 2x^2 - color(red)(5x) - color(blue)(18)#
#5x - color(red)(5x) + 18 - color(blue)(18) = 2x^2 - color(red)(5x) - color(blue)(18)#
#0 + 0 = 2x^2 - 5x - 18#
#0 = 2x^2 - 5x - 18#
We can now factor this.
#2x^2 - 5x - 18 = 0#
#(2x - 9)(x + 2) = 0#
We can now solve both of these for #0#:
#2x - 9 = 0#
#2x - 9 + 9 = 0 + 9#
#2x - 0 = 9#
#2x = 9#
#(2x)/2 = 9/2#
#x = 9/2#
and
#x + 2 = 0#
#x + 2 - 2 = 0 - 2#
#x + 0 = -2#
#x = -2#
Because you can't produce NEGATIVE units the answer is 9/2 or 4.5 units are needed to be produced to break even.
But because you can't create a 1/2 a unit you need to round up to 5 units.
