Question

express each of following in sum of a perfect square x²+y²+z²+XY+yz-zx​

Answer 1

Question:

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

Solution:

• 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 .