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

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Solution

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

$1) {(3 a - 7 b - c)}^{2}$

$a = 3 a, b = - 7 b, c = - c$ [comparing with identity(i)]

$= {(3 a)}^{2} + {(- 7 b)}^{2} + {(- c)}^{2} + 2 (3 a) (- 7 b) + 2 (- 7 b) (- c) + 2 (- c) (3 a)$

$= {9 a}^{2} + 49 b^{2} + c^{2} - 42 a b + 14 b c - 6 a c$

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

$a = x, b = 2 y, x = 4 z$ [comparing with identity(i)]

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

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

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