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Question

If a,b and c are real numbers and a2+b2+c2abbcca=0 then show that a=b=c?

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Solution

Consider a2+b2+c2abbcca=0
Multiply both sides with 2, we get
2(a2+b2+c2abbcca)=0
2a2+2b2+2c22ab2bc2ca=0
a2+a2+b2+b2+c2+c22ab2bc2ca=0
(a22ab+b2)+(b22bc+c2)+(c22ca+a2)=0
(ab)2+(bc)2+(ca)2=0
Since the sum of squares is zero, it means each term should be zero.
(ab)2=0,(bc)2=0,(ca)2=0
ab=0,bc=0,ca=0
a=b,b=c,c=a
a=b=c
Hence proved.


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