Question 28
Expand the following
(i) $(4 a - b + 2 c)^{2}$

(ii) $(3 a - 5 b - c)^{2}$

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

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Solution

(i)
$(4 a - b + 2 c)^{2} = (4 a^{2}) + (- b)^{2} + (2 c)^{2} + 2 (4 a) (- b) + 2 (- b) (2 c) + 2 (2 c) (4 a)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= 16 a^{2} + b^{2} + 4 c^{2} - 8 a b - 4 b c + 16 a c$

(ii)
$(3 a - 5 b - c)^{2} = (3 a)^{2} + (- 5 b)^{2} + (- c)^{2} + 2 (3 a) (- 5 b) + 2 (5 b) (- c) + 2 (- c) (3 a)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= 9 a^{2} + 25 b^{2} + c^{2} - 30 a b - 10 b c - 6 a c$

(iii)
$(- x + 2 y - 3 z)^{2} = (- x)^{2} + (2 y)^{2} + (- 3 z)^{2} + 2 (- x) (2 y) + 2 (2 y) (- 3 z) + 2 (- 3 z) (- x)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= x^{2} + 4 y^{2} + 9 z^{2} - 4 x y - 12 y z + 6 x z$

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Question 28
Expand the following
(i) $(4 a - b + 2 c)^{2}$

(ii) $(3 a - 5 b - c)^{2}$

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

$3 a - [4 a - ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 3 a - 5 b - 3$ { $2 a - (3 a - ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 2 a - 3 b)$ } $]$ equals

Expand each of the following, using suitable identities:

(i) (ii)

(iii) (iv)

(v) (vi)

a = 2^i + 5^j - 7^k, b =^i + 3^j - 5^k

(3 a - 5 b) . (4 a \times 5 b)

3 a + 2 b = 24

4 a + 5 b = 53

a + b

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