Simplify:$(a + b + c)^2 + (a - b + c)^2 + (a + b - c)^2$

Given:

$(a + b + c)^2 +   (a - b + c)^2 + (a + b - c)^2$

To do:

We have to simplify $(a + b + c)^2 +   (a - b + c)^2 + (a + b - c)^2$.

Solution:

We know that,

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

Therefore,

$(a+b+c)^{2}+(a-b+c)^{2}+(a+b-c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+a^{2}+b^{2}+c^{2}-2 a b-2 b c+2 c a+a^{2}+b^{2}+c^{2}+2 a b-2 b c-2 c a$

$=3 a^{2}+3 b^{2}+3 c^{2}+2 a b-2 b c+2 c a $

$=3(a^{2}+b^{2}+c^{2})+2(a b-b c+c a)$

Hence, $(a + b + c)^2 +   (a - b + c)^2 + (a + b - c)^2=3(a^{2}+b^{2}+c^{2})+2(a b-b c+c a)$.

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Updated on: 10-Oct-2022

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