# Prove that:  x2 + y2+ z2 - xy - yz - zx  is always positive for distinct values of x, y and z.

Dear Student,

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 + 2y+ 2z2 - 2xy - 2yz - 2zx

$⇒$ x2x2y2  y2  z2 + z2 - 2xy - 2yz - 2zx

$⇒$ x2y2  - 2xy  + y2  z2 - 2yz + z2x2 - 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 .