The break-even point is when revenue, R(h) is the same as cost, C(h).
R(h)=C(h)
220h-160 = 20h²-400
Gather all the variables on one side by subtracting 220h:
220h-160-220h = 20h²-400-220h
-160=20h²-220h-400 (we can move these around so long as we take their respective signs with them)
We want our quadratic equation to equal 0 to solve it, so add 160 to both sides:
-160+160=20h²-220h-400+160
0=20h²-220h-240
All 3 terms of this quadratic are divisible by 20, we can factor 20 out:
0 = 20(h² - 11h - 12)
The quadratic that is left is easily factorable. We want factors of -12 that sum to -11; that would give us -12*1, because -12+1 = -11. Thus we have
0 = 20(h - 12)(h + 1)
Using the zero product property, we know that one of the factors must be 0 in order for the product to be 0. 20 ≠ 0, so it must be either h-12 or h+1:
h-12 = 0
Add 12 to both sides:
h - 12 + 12 = 0 + 12
h = 12
h+1 = 0
Subtract 1 from both sides:
h + 1 - 1 = 0 - 1
h = -1
Since a negative number of hours is not realistic, the answer must be h = 12 hours.
