Two numbers have a sum of 20. Determine the lowest possible sum of their squares.

Two numbers have a sum of 20. Determine the lowest possible sum of their squares.


If sum of two numbers is 20, let one number be x. Then the other number would be 20 - x.

The sum of their squares is:
x^2+(20 - x)^2

Expand this and we get:
x^2 + 400 - 40x + x^2

Combine like terms:
2x^2 - 40x + 400

Rewrite this:
2(x^2 - 20x + 100 - 100) + 400
2(x - 10)^2 - 200 + 400
2(x−10)^2 + 200

The sum of squares of two numbers is sum of two positive numbers, one of which is a constant of 200.

The other number, 2(x - 10)^2, can change according to the value of x. The least value could be 0, when x=10

Therefore, the minimum value of sum of squares of two numbers is 0 + 200 = 200 when x = 10.

If x = 10, then the other number is 20 - 10 = 10.