# Thread: Finding two unknown numbers

1. ## Finding two unknown numbers

Determine two numbers whose difference is -4 and whose product is a minimum. Let x be one number and let y be the other.

There is another question similar to this but with maximum instead of minimum, the concept should be the same, right?

2. Originally Posted by brownjovi
Determine two numbers whose difference is -4 and whose product is a minimum. Let x be one number and let y be the other.

There is another question similar to this but with maximum instead of minimum, the concept should be the same, right?
x - y = -4

x = y - 4

P = xy = (y-4)y

what value of y makes P a minimum?

3. Originally Posted by skeeter
x - y = -4

x = y - 4

P = xy = (y-4)y

what value of y makes P a minimum?
Is it two?

4. I am a newbie myself, but I think that one way to solve this is to think something along these lines.

If y is negative, the product y(y-4) will be positive.
If y is positive, there will be a few numbers for which the product y(y-4) is negative, and some for which the product will be positive.

In this particular case it is easy to manually figure out for which values of y the product will be negative, since there are only so many positive values of y for which (y-4) will remain negative.

5. If we consider $ax^2+bx+c$ completing the square gives $a\left(x-\tfrac{b}{2a}\right)^2+c-\frac{b^2}{4a}$ which when $a>0$ we may clearly see that $a\left(x-\tfrac{b}{2a}\right)^2+c-\frac{b^2}{4a}\geqslant c-\frac{b^2}{4a}$ so that it attains it's minimum at $x=\frac{b}{2a}$. Geometrically all we did was find the vertex of the parabola.

6. Originally Posted by Sabo
I am a newbie myself, but I think that one way to solve this is to think something along these lines.

If y is negative, the product y(y-4) will be positive.
If y is positive, there will be a few numbers for which the product y(y-4) is negative, and some for which the product will be positive.
Actually, no. There will be an infinite wet of numbers for which the product y(y- 4) is negative. You cannot assume the number will be a whole number. Completing the square, as Drexel28 suggested, will always give you the minimum or maximum point (vertex of the parabola) of a quadratic function.

In this particular case it is easy to manually figure out for which values of y the product will be negative, since there are only so many positive values of y for which (y-4) will remain negative.
Again, no, there are an infinite number of them.

7. Right. Stuck in integer land. =/