SOLUTION: Hi I have a word problem with a linear equation of f(x)=-0.002x^2+1.4x-400. X is the number of pretzels sold, and I was told to find a max profit. How would I start this equation?

Question 1028396: Hi I have a word problem with a linear equation of f(x)=-0.002x^2+1.4x-400. X is the number of pretzels sold, and I was told to find a max profit. How would I start this equation?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this is a quadratic equation.

set the equation equal to 0 and you get -.002x^2 + 1.4x - 400 = 0

the equation is now in standard quadratic equation form of ax^2 + bx + c = 0

a is the coefficient of the x^2 term which is -.002
b is the coefficient of the x term which is 1.4
c is the constant term which is -400

the x-coordinate of max/min point of a quadratic equation can be found using the following formula:

x = -b/2a.

in your problem, this becomes x = -1.4 / (2*(-.002)) which becomes -1.4 / -.004 which results in x = 350.

when x = 350, the equation of f(x) = -.002x^2 + 1.4x - 400 becomes f(350) = -.002 * (350^2 + 1.4 * 350 - 400 which results in f(350) = -155.

that's a negative number which indicates that you didn't make a profit, but took a loss, with the minimum loss occurring when you sold 350 pretzels.

this assumes that f(x) is supposed to represent your profit.

the graph of your equation is shown below.

this graph confirms the manual calculation using the max/min formula.

when the coefficient of the x^2 term of a quadratic equation is negative, the max/min formula give you the max.

$$$

in the graph, y means the same thing as f(x).

the x-axis tells you the number of pretzels sold.
the y-axis tells you the result of the equation using that value of x.

if f(x) is supposed to represent your profit, then you didn't make any profit, but took a loss instead.

you can say that your maximum profit is when 350 pretzels are sold.

you can say that your maximum profit is -155.

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