Mathematics
If x, y and z are three different numbers, then prove that :
x2 + y2 + z2 - xy - yz - zx is always positive.
Expansions
27 Likes
Answer
Given,
⇒ x2 + y2 + z2 - xy - yz - zx
Multiplying the above equation by 2,
⇒ 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
From above equation we can see that for distinct value of x, y and z given equation is always positive.
Answered By
18 Likes