Revenue = cost

30000+45x+0.3x^2 = 605x-0.7x^2

x^2-560x+30000 =0

This equation is of form ax^2+bx+c=0

a = 1 b = -560 c = 30000

x=[-b+/-sqrt(b^2-4ac)]/2a]

x=[560 +/-sqrt(-560^2-4(1)(30000)]/(2)(1)

discriminant is b^2-4ac =193600

x=[560 +√(193600)] / (2)(1)

x=[560 -√(193600)] / (2)(1)

x=[560+440] / 2

x=[560-440] / 2

The roots are 500 and 60

x=60 and x=500 are break-even points.

At breakeven point total revenue equals the total cost:

C(x) = R(x)

30000 + 45x + 0.3x^2 = 605x - 0.7x^2

x^2 - 560x + 30000 = 0

(x - 60)(x - 500) = 0

x = 60, 500