Question

The sum of two numbers is 20. Find the minimum possible sum of their squares?

Answer 1

`10+10 = 20`
`10^2 +10^2=200`.

Explanation:

`a+b=20`
`a^2 + b^2= x`
For `a` and `b`:
`1^2+19^2=362`
`2^2+18^2=328`
`3^2+17^2=298`
From this, you can see that closer values of `a` and `b` will have a smaller sum. Thus, for `a=b`, `10+10 = 20` and `10^2 +10^2=200`.

Answer 2

Minimum value of sum of squares of two numbers is `200`, which is when both numbers are `10`

Explanation:

If sum of two numbers is `20`,

let one number be `x` and then other number would be `20-x`

Hence their sum of squares is

`x^2+(20-x)^2`

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

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

= `2(x^2-20x+100-100)+400`

= `2(x-10)^2-200+400`

= `2(x-10)^2+200`

Observe that the sum of squares of two numbers is sum of two positive numbers, one of whom is a constant i.e. `200`

and other `2(x-10)^2`, which can change according to value of `x` and its least value could be `0`, when `x=10`

Hence minimum value of sum of squares of two numbers is `0+200=200`, which is when `x=10`, which is when both numbers are `10`.