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Break-even value is when the cost of production equals the revenue earned. No profit is earned.
Therefore, you need to set the production cost = revenue earned. Defining the variable, x=products produced.
x^2-8x=20x+60. Solve for x.
Factoring the quadratic, what 2 numbers can you multiply together to get -60 and add together to get -28? Answer: +2 and -30.
Using the zero product property, set each factor to 0 and solve.
x=30 and x=-2. As -2 is extraneous (can't have -2 products), the answer is x=30 products.
The break-even point is when revenue earned is equal to the cost incurred.
For the production of x, the revenue is given by x^2 - 8x and the cost is 20x + 60. Equating the two and solving for x:
x^2 - 8x = 20x + 60
=> x^2 - 28x - 60 = 0
=> x^2 - 30x + 2x - 60 = 0
=> x(x - 30) + 2(x - 30) = 0
=> (x + 2)(x - 30) = 0
=> x = -2 and x = 30
Assuming x can only take on positive values, the break-even value of x is 30
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