Simplify $l^{2} + m^{2} + m^{2} + n^{2} + n^{2} + l^{2} + 2 l m + 2 m n + 2 n l$ .

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Solution

We have, $l^{2} + m^{2} + m^{2} + n^{2} + n^{2} + l^{2} + 2 l m + 2 m n + 2 n l$

$= 2 l^{2} + 2 m^{2} + 2 l m + 2 n^{2} + 2 m n + 2 n l$ [After re-arranging]

$= 2 {l^{2} + m^{2} + n^{2} + l m + m n + n l}$

OR,
$l^{2} + m^{2} + m^{2} + n^{2} + n^{2} + l^{2} + 2 l m + 2 m n + 2 n l$
$= [l^{2} + m^{2} + 2 m n] + [m^{2} + n^{2} + 2 m n] + [n^{2} + l^{2} + 2 n l]$
[Applying the identity $(a^{2} + b^{2} + 2 a b) = (a + b)^{2}$ ]
$= (l + m)^{2} + (m + n)^{2} + (n + l)^{2}$

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Add the following.

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