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if-a-b-c-2-a-b-2-c-2-a-2-b-c-2-0-then-prove-that-a-c-or-a-c-2b-




Question Number 42009 by 737954 last updated on 16/Aug/18
if (a+bω+cω^2 )+(aω+bω^2 +c)^2 +(aω^2 +b+cω)^2 =0  then prove that a=c  or   a+c=2b
if(a+bω+cω2)+(aω+bω2+c)2+(aω2+b+cω)2=0thenprovethata=cora+c=2b
Answered by tanmay.chaudhury50@gmail.com last updated on 16/Aug/18
a^2 +a^2 w^2 +a^2 w^4 +b^2 w^2 +b^2 w^4 +b^2 +c^2 w^4 +c^2 +c^2 w^2   +2abw+2abw^3 +2abw^2 +2bcw^3 +2bcw^2 +2bcw  +2acw^2 +2acw+2acw^3   =a^2 (1+w^2 +w)+b^2 (w^2 +w+1)+c^2 (w+1+w^2 )+  2ab(w+w^3 +w^2 )+2bc(w^3 +w^2 +w)+2ac(w^2 +w+w^3 )  a×0+b×0+c×0+(1+w+w^2 )(2ab+2bc+2ac)  =0×(2ab+2bc+2ac  =0   pls check the question  because given expression=0 on simplification  but can not prove a=c ora+c=2b    w^4 =w^3 .w=1.w=w  1+w+w^2 =0   formula    i have squared and then added identical terms  by mind calculation  becauze  (A+B+C)^2 =A^2 +B^ +C^2 +2AB+2BC+2AC
a2+a2w2+a2w4+b2w2+b2w4+b2+c2w4+c2+c2w2+2abw+2abw3+2abw2+2bcw3+2bcw2+2bcw+2acw2+2acw+2acw3=a2(1+w2+w)+b2(w2+w+1)+c2(w+1+w2)+2ab(w+w3+w2)+2bc(w3+w2+w)+2ac(w2+w+w3)a×0+b×0+c×0+(1+w+w2)(2ab+2bc+2ac)=0×(2ab+2bc+2ac=0plscheckthequestionbecausegivenexpression=0onsimplificationbutcannotprovea=cora+c=2bw4=w3.w=1.w=w1+w+w2=0formulaihavesquaredandthenaddedidenticaltermsbymindcalculationbecauze(A+B+C)2=A2+B+C2+2AB+2BC+2AC

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