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Expansion of (a ± b ± c)²

Simplify: 2...

Question

Simplify: ${(2 m - n - 3 p)}^{2} + 4 m n - 6 n p + 12 p m$

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Solution

Using the identity is $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$ .

Given: ${(2 m - n - 3 p)}^{2} + 4 m n - 6 n p + 12 p m$ and apply the above identity as follows:

${(2 m - n - 3 p)}^{2} + 4 m n - 6 n p + 12 p m$
$= {[{(2 m)}^{2} + {(- n)}^{2} + {(- 3 p)}^{2} + (2 \times 2 m \times - n) + (2 \times - n \times - 3 p) + (2 \times - 3 p \times 2 m)]}^{2} + 4 m n - 6 n p + 12 p m$
$= 4 m^{2} + n^{2} + 9 p^{2} - 4 m n + 6 n p - 12 p m + 4 m n - 6 n p + 12 p m = 4 m^{2} + n^{2} + 9 p^{2}$

Hence, ${(2 m - n - 3 p)}^{2} + 4 m n - 6 n p + 12 p m = 4 m^{2} + n^{2} + 9 p^{2}$ .

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