Question

Two numbers have a difference of 20. How do you find the numbers if the sum of their squares is a minimum?

Answer 1

`-10,10`

Explanation:

Two numbers `n,m` such that `n-m=20`

The sum of their squares is given by

`S=n^2+m^2` but `m = n-20` so

`S=n^2+(n-20)^2=2n^2-40n+400`

As we can see, `S(n)` is a parabola with a minimum at

`d/(dn)S(n_0) = 4n_0-40=0` or at `n_0 = 10`

The numbers are

`n=10, m=n-20=-10`

Answer 2

10 and -10

Solved without Calculus.

Explanation:

In Cesareo’s answer `d/(dn)S(n_0)` is Calculus. Let’s see if we can solve this without calculus.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(magenta)("Let the first number be "x)`
Let the second number be `x+20`

Set `" "y=x^2+(x+20)^2`

`y=x^2+x^2+40x+400`

`y=2x^2+40x+400 larr" "y" is the sum of their squares"`

`color(red)("So we need to find the value of x that gives the minimum value")` `color(red)("of "y)`

This equation is a quadratic and as the `x^2` term is positive then it its general shape is of form `uu`. Thus the vertex is the minimum value for `y`

Write as `y=2(x^2+20x)+400`

What follows is part of the process for completing the square.

Consider the 20 from `20x`

`color(magenta)("Then the first number is: "x_("vertex")=(-1/2)xx20= -10)`

Thus the first number is `x=-10`
The second number is `" "x+20 =-10+20 =10`

`" "color(green)(bar(ul(|color(white)(2/2)"The two numbers are: -10 and 10 "|)))`