Question

Prove that x^2+y^2+z^2-xy-yz-zx is always posotive

Answer 1

( x^2 + y^2 + z^2 - xy - yz - xz ) × 2 / 2
= 1/2 ( 2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2xz )
= 1 /2 ( x^2 + x^2 + y^2 + y^2 + z^2 + z^2 - 2xy -2yz - 2xz )
On rearranging the terms we get ,
= 1/2 {( x^2 + y^2 - 2xy) + (y^2 + z^2 - 2yz) + ( x^2 + z^2 - 2xz) }
= 1/2 { ( x - y )^2 + ( y -z )^2 + ( x - z )^2 }
In above expression ,
1/2 > 0
( x - y )^2 > 0
( y - z )^2 >0 and
( x - z )^2 >0

=> Given expression is always Positive.