Question

Find two numbers whose difference is 164 and whose product is a minimum. (smaller number) (larger number)

Answer 1

The lowest possible product would be -6724 given the numbers 82 and -82. 

We can find this by setting the first number as x + 164. The other number would have to be simply x since it has to have a 164 difference. 

Next we'll multiply the numbers together. 
x(x+164)
x^2 + 164x

Now we want to minimize this as much as possible, so we'll find the vertex of this quadratic graph. You can do this by finding the x value as -b/2a, where b is the number attached to x and a is the number attached to x^2

-b/2a = -164/2(1) = -164/2 = -82

So we know one of the values is -82. We can plug that into the equation to find the second. 

x + 164
-82 + 164 
82