Question

a2 + b2 + c2 -ab - bc - ca=0, Prove that a =b=c.​

Answer 1

Answer:

a+b+c=0 --- 1

-a=b+c−a=b+c From 1 --- 2

-b=c+a−b=c+a From 1 --- 3

-c=a+b−c=a+b From 1 --- 4

Substitute into a^2(b+c)+b^2(c+a)+c^2(a+b)+3abc=0a

2

(b+c)+b

2

(c+a)+c

2

(a+b)+3abc=0 .

-a^3-b^3-c^3+3abc=0−a

3

−b

3

−c

3

+3abc=0

a^3+b^3+c^3-3abc=0a

3

+b

3

+c

3

−3abc=0

(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0(a+b+c)(a

2

+b

2

+c

2

−ab−bc−ca)=0

0\times (a^2+b^2+c^2-ab-bc-ca)=00×(a

2

+b

2

+c

2

−ab−bc−ca)=0 From 1

Hence shown.

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