Question

The profit function for a product is given by p(x)=-x^3+3x^2+900x-2100, where x is the number of units produced and sold and P is in hundreds of dollars. Use factoring by grouping to find the numbers that will give a profit of $60,000.

Answer 1

x = 30 and x = 3 both solve the equation correctly. See work below.

Explanation:

The very first step is to realize that p(x) is in hundreds of dollars and that the given number is not. 60000 is not in hundreds of dollars. It is in dollars. 60000 dollars = 60000/100 = 600 hundreds of dollars.

Make the polynomial = 600

-x^3+3x^2+900x-2100 = 600 Subtract 600 from both sides

-x^3+3x^2+900x-2100 - 600 = 0 Combine

-x^3+3x^2+900x- 2700 = 0 Group the first and second term and the Thrid and fourth terms.

(-x^2 + 3x^2) + (900x - 2700) = 0 Bring out the common factors from both groups.

-x^2(x - 3) + 900(x - 3) = 0 Factor out x - 3 on both sides of the plus sign

(- x^2 + 900)(x - 3) = 0 Reverse -x^2 + 900 to 900 - x^2

(900 - x^2)(x - 3) = 0 factor (900 - x^2) using the difference of squares

(30 - x)(30 + x) (x - 3) = 0

The two answers that work are

30 - x = 0 Solve

- x = - 30 Divide by - 1

-x/-1 = -30/-1 Combine

x = 30

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x - 3 = 0

x = 3

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There are two values that will work to give p(x) = 600. There is no way to account for why both values work.

The graph below confirms both answers.