An electronics store is deciding how to price one of its products. the equation p=-25d^2+500d predicts the total profit p as a function of the product's price in dollars d. what price will produce the highest total profit?

1. The function calculating the profit in terms of the price in dollars d, is the quadratic function , whose graph is a concave parabola, that is a parabola which opens downwards, because the coefficent of  is negative. 

2. The function P takes its maximum value at the vertex, so we want to find the x coordinate of the vertex. 

3.  is the factorized form of the function P, which clearly gives the roots (solutions) 0 and 20 of P(d)=0, which are the x intercepts of the parabola. 

4. The axis of symmetry, which contains the vertex, is the vertical line passing through the point equidistant to 0 and 20, that is 10.

5. Remark : The vertex is (10, P(10))=(10, -25*10(10-20))=(10, 2500)

So for price d=10 dollars, the (expected) Profit is 2500 dollars.

6. Remark 2. The x coordinate is also found by the formula -b/2a, where b is the coefficient of the linear term, and a the coefficient of the quadratic term of a quadratic expression. So in our case -b/2a=-500/2(-25)=-500/-50=10

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