Question 1064107
let x be the larger number
the smaller number is therefore x-6.


the difference between the numbers will always be 6.


their product would be x * (x-6) = x^2 - 6x.


let y be their product and you get y = x^2 - 6x.


you can use the max/min value formula to get the value of x where y is either max or min.


since the coefficient of the x^2 term is positive, the max/min value will be min.


set y = 0 and the quadratic equation is in standard form of ax^2 + bx = 0


this means that a = 1 and b = 6.


the min/max formula for x is x = -b/2a.


you will get x = -b/2a which becomes x = 6/2 = 3.


when x = 3, the value of y is 9 - 18 = -9.


the minimum value of the product is therefore -9.


here is a graph of the quadratic equation in standard form.


{{{graph(500,500,-12,12,-12,12,x^2 - 6x)}}}