Prove that a2+b2+c2-ab-bc-ca is always non-negative for all values of a,b and c

CBSE Class 9 CBSE Class 9 Maths
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1

Prove that a2+b2+c2-ab-bc-ca is always non-negative for all values of a,b and c.

2

a2 + b2 + c2 - ab - bc - ca
= (1/2)(2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca)
= (1/2)(a2 - 2ab + b2 + b2- 2bc + c2 + c2 - 2ca + a2)
= (1/2)[(a-b)2+ (b-c)2+ (c-a)2]
Now we know that, the square of a number is always greater than or equal to zero.
Hence, (1/2)[(a-b)2 + (b-c)2+ (c-a)2] ? 0 => it is always non-negative.