Math, asked by azizkm786, 1 year ago

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

Answers

Answered by TheUrvashi
93
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.


Answered by arpan7de
81

Answer:

Hello

We have : x2 + y2  + z2 - xy - yz - zy and we have to prove that it is always positive.

Now we  multiply the whole equation by 2 and get :

2 ( x² + y²  + z² - xy - yz - zx  )

⇒2x² + 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

Step-by-step explanation:

From above equation we can conclude that thegiven equation is always positive because the square of any rational number is always positive

Hope you won't face any more doubt after understanding the above mentioned information.

Thanks....

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