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Question
Simplify (a + b + c)2 + (a - b + c)2 + (a + b - c)2
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Solution
We have
(a + b + c)2 + (a - b + c)2 + (a + b - c)2
`= [a^2 + b^2 + c^2 + 2ab + 2bc + 2ca] + [a^2 + b^2 + c^2 - 2bc - 2ab + 2ca] + [a^2 + b^2 + c^2 - 2ca - 2bc + 2ab]`
`[∵ (x + y + z)^2 = x^2 + y^2 + 2xy + 2yz + 2zx]`
`= 3a^2 + 3b^2 + 3c^2 + 2ab + 2bc + 2ca - 2bc - 2ab + 2ca - 2ca - 2bc + 2ab`
`= 3a^2 + 3b^2 + 3c^2 + 2ab - 2bc + 2ca`
`= 3(a^2 + b^2 + c^2) + 2(ab - bc + ca)`
`∴(a + b + c)^2 + (a - b + c)^2 + (a + b - c)^2 = 3(a^2 + b^2 + c^2) + 2[ab - bc + ca]`
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