Expand each of the following, using suitable identities:
$1) {(x + 2 y + 4 z)}^{2}$
$2) {(3 x - 7 y - z)}^{2}$

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Solution

${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 a c$ ........(1)

$1) {(x + 2 y + 4 z)}^{2}$
$a = x, b = 2 y, c = 4 z$
Substituting the values of a, b and c in (1)
${(x + 2 y + 4 z)}^{2}$
$= x^{2} + {(2 y)}^{2} + {(4 z)}^{2} + 2 (x) (2 y) + 2 (2 y) (4 z) + 2 (4 z) (x)$

$= {(x)}^{2} + 4 {(y)}^{2} + 16 {(z)}^{2} + 4 x y + 16 y z + 8 z x$

$2) {(3 x - 7 y - z)}^{2}$
$a = 3 x, b = - 7 y, c = - z$
Substituting the values of a, b and c in (1)

${(3 x - 7 y - z)}^{2}$
$= {(3 x)}^{2} + {(- 7 y)}^{2} + {(- z)}^{2} + 2 (3 x) (- 7 y) + 2 (- 7 y) (- z) + 2 (- z) (3 x)$

$= 9 x^{2} + 49 y^{2} + z^{2} - 42 x y + 14 y z - 6 x z$

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