An entrepreneur estimates that the total profit (profit = total revenue minus total cost) from his proposed company will be given by the function P(x) = x^3 – 4x^2 + 3x – 12, where P is in hundreds of dollars and x is the number of years elapsed after the company starts operations. In how many years (x) will the company break even (no profit, no loss)?
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would it be 4
x can not be negative since it is time in years. so x-4 = 0 gives x=4; hence 4 years will bring in breakeven point
So we just need to find the roots of that expression
\[\large P(x) = x^3 – 4x^2 + 3x – 12\]
So we can factor this one easily by grouping
\[\large 0 = (x^3 - 4x^2) + (3x - 12)\]
Factor out an x^2 from the first...and factor out a 3 from the second
\[\large x^2(x - 4) + 3(x - 4) = 0\]
Combine the common factors and put the rest together
\[\large (x-4)(x^2 + 3) = 0\]
So solve that for x to make it equal 0
and as you just posted above...it is indeed 4 years :)