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

2 Answers
Apr 4, 2018

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.

Apr 4, 2018

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.