X 2 y 2 z 2 xy yz zx
Asked by admin @ in Math viewed by 305 People
Prove that x^2 +y^2 +z^2-xy-yz-zx is always positive.
Answered by admin @
Hey
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 .
Hope this information will clear your doubts about topic.
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