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if-bc-2-ca-2-ab-2-a-2-b-b-2-c-c-2-a-0-prove-that-a-b-b-c-c-a-

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Question Number 171204 by mokys last updated on 09/Jun/22
if : bc^2 +ca^2 +ab^2 −a^2 b−b^2 c−c^2 a=0 prove that :a = b , b = c , c = a
if:bc2+ca2+ab2−a2b−b2c−c2a=0provethat:a=b,b=c,c=a
Answered by som(math1967) last updated on 10/Jun/22
ab^2 −a^2 b+bc^2 −c^2 a+ca^2 −b^2 c=0 ⇒−ab(a−b)−c^2 (a−b)+c(a^2 −b^2 )=0 ⇒(a−b){−ab−c^2 +c(a+b)}=0 ⇒(a−b)(bc−c^2 −ab+ca)=0 ⇒(a−b){c(b−c)−a(b−c)}=0 ⇒(a−b)(b−c)(c−a)=0 ∴a=b,b=c,c=a
ab2−a2b+bc2−c2a+ca2−b2c=0⇒−ab(a−b)−c2(a−b)+c(a2−b2)=0⇒(a−b){−ab−c2+c(a+b)}=0⇒(a−b)(bc−c2−ab+ca)=0⇒(a−b){c(b−c)−a(b−c)}=0⇒(a−b)(b−c)(c−a)=0∴a=b,b=c,c=a

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