Expand the following, using suitable identities: ${(- 2 x + 3 y + 2 z)}^{2}$

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Solution

Use appropriate algebraic identity and simplify given expression
Given ${(- 2 x + 3 y + 2 z)}^{2}$
Using identity,
${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
By comparing ${(- 2 x + 3 y + 2 z)}^{2}$ with ${(a + b + c)}^{2}$ , we get
$a = - 2 x, b = 3 y$ and $c = 2 z$
$\begin{array}{rcl} ∴ {(- 2 x + 3 y + 2 z)}^{2} & = & {(- 2 x)}^{2} + {(3 y)}^{2} + {(2 z)}^{2} + 2 (- 2 x) (3 y) + 2 (3 y) (2 z) + 2 (2 z) (- 2 x) \\ = & 4 x^{2} + 9 y^{2} + 4 z^{2} - 12 x y + 12 y z - 8 z x \end{array}$
Hence, ${(- 2 x + 3 y + 2 z)}^{2} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 4 \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} \end{array}$

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