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Question
Prove that : x2+ y2 + z2 - xy - yz - zx is always positive.
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Solution
x2 + y2 + z2 - xy - yz - zx
= 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) + (z2 + x2 - 2zx) + (y2 + z2 - 2yz)
= (x - y)2 + (z - x)2 + (y - z)2
Since square of any number is positive, the given equation is always positive.
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