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Expand each of the following, using suitable identities: (i) (

Question

Expand each of the following, using suitable identities:
(i) $(x + 2 y + 4 z)^{2}$
(ii) $(2 x - y + z)^{2}$
(iii) $(- 2 x + 3 y + 2 z)^{2}$
(iv) $(3 a - 7 b - c)^{2}$
(v) $(- 2 x + 5 y - 3 z)^{2}$
(vi) $[\frac{1}{4} a - \frac{1}{2} b + 1]^{2}$

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Expand each of the following, using suitable identities:
(i)

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

(ii)

(2 x - y + z)^{2}

(iii)

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

(iv)

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

(v)

(- 2 x + 5 y - 3 z)^{2}

(vi)

[\frac{1}{4} a - \frac{1}{2} b + 1]^{2}

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Text solutionVerified

We know,

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 ab + 2 b c + 2 a c

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

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

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

ii)

(2 x - y + z)^{2}

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

= 4 x^{2} + y^{2} + z^{2} - 4 x y - 2 yz + 4 x z

iii)

(- 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 x) (2 z)

= 4 x^{2} + 9 y^{2} + 4 z^{2} - 12 x y + 12 yz - 8 x z

iv)

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

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

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

(- 2 x + 5 y - 3 z)^{2}

= (- 2 x)^{2} + (5 y)^{2} + (- 3 z)^{2} + 2 (- 2 x) (5 y) + 2 (5 y) (- 3 z) + 2 (- 3 z) (- 2 x)

= 4 x^{2} + 25 y^{2} + 9 z^{2} - 20 x y - 30 yz + 12 x z

iv)

(\frac{1}{4} a - \frac{1}{2} b + 1)^{2}

= (\frac{1}{4} a)^{2} + (- \frac{1}{2} b)^{2} + (1)^{2} + 2 (\frac{1}{4} a \times - \frac{1}{2} b) + 2 (- \frac{1}{2} b \times 1) + 2 (\frac{1}{4} a \times 1)

= \frac{1}{16} a^{2} + \frac{1}{4} b^{2} + 1 - \frac{1}{4} ab - b + \frac{1}{2} a

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Question Text	Expand each of the following, using suitable identities: (i) $(x + 2 y + 4 z)^{2}$ (ii) $(2 x - y + z)^{2}$ (iii) $(- 2 x + 3 y + 2 z)^{2}$ (iv) $(3 a - 7 b - c)^{2}$ (v) $(- 2 x + 5 y - 3 z)^{2}$ (vi) $[\frac{1}{4} a - \frac{1}{2} b + 1]^{2}$
Answer Type	Text solution:1
Upvotes	150