Question

Prove that:

x² + y² + z² - xy - yz - zx is always positive.

Answer 1

We have : x2 + y2 + z2 - xy - yz - zy

Now we multiply by in whole equation and get

2 ( x2 + y2 + z2 - xy - yz - zx )

⇒2 x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx

⇒ x2 + x2 + y2 + y2 + z2 + z2 - 2xy - 2yz - 2zx

⇒ x2 + y2 - 2xy + y2 + z2 - 2yz + z2 + x2 - 2zx

⇒ ( x - y )2 + ( y - z )2 + ( z - x )2

From above equation we can see that for distinct value of x , y and z given equation is always positive .