Skylar repairs microwaves. her revenue equals r(h) = 80 + 30h dollars for every h hours she spends repairing microwaves. her overhead cost equals c(h) = 10h^2 - 20 dollars.
after how many hours does she break even?

Given:
R(h) = 80 + 30h
C(h) = 10h² - 20

Break even is when the revenue and overhead cost is equal and profit is 0.

R(h) - C(h) = 0

80 + 30h - (10h² - 20) = 0
80 + 30h - 10h² + 20 = 0
-10h² + 30h + 100 = 0
-10(h² - 3h -10) = 0

h² - 3h - 10) = 0
(h - 5)(h + 2) = 0
h(h+2) - 5(h+2) ⇒ h² + 2h - 5h - 10 = h² - 3h - 10

h - 5 = 0 ; h = 5
h + 2 = 0 ; h = -2

She breaks even at 5 hours. There is no negative hours so 5 is the correct answer.

R(h) = C(h)
80 + 30h = 10h² - 20
80 + 30(5) = 10(5²) - 20
80 + 150 = 10(25) - 20
230 = 250 - 20
230 = 230