Question 29
Factorise the following:
(i) $9 x^{2} + 4 y^{2} + 16 z^{2} + 12 x y - 16 y z - 24 x z$

(ii) $25 x^{2} + 16 y^{2} + 4 z^{2} - 40 x y + 16 y z - 20 x z$

(iii) $16 x^{2} + 4 y^{2} + 9 z^{2} - 16 x y - 12 y z + 24 x z$

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Solution

(i) $9 x^{2} + 4 y^{2} + 16 z^{2} + 12 x y - 16 y z - 24 x z$
$= (3 x)^{2} + (2 y)^{2} + (- 4 z)^{2} + 2 (3 x) (2 y) + 2 (2 y) (- 4 z) + 2 (- 4 z) (3 x)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= (3 x + 2 y - 4 z)^{2} = (3 x + 2 y - 4 z) (3 x + 2 y - 4 z)$

(ii) $25 x^{2} + 16 y^{2} + 4 z^{2} - 40 x y + 16 y z - 20 x z$
$= (- 5 x)^{2} + (4 y)^{2} + (2 z)^{2} + 2 (- 5 x) (4 y) + 2 (4 y) (2 z) + 2 (2 z) (- 5 x)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= (- 5 x + 4 y + 2 z)^{2} = (- 5 x + 4 y + 2 z) (- 5 x + 4 y + 2 z)$

(iii) $16 x^{2} + 4 y^{2} + 9 z^{2} - 16 x y - 12 y z + 24 x z$
$= (4 x)^{2} + (- 2 y)^{2} + (3 z)^{2} + 2 (4 x) (- 2 y) + 2 (- 2 y) (3 z) + 2 (4 x) (3 z)$
$[U s i n g t h e i d e n t i t y, (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a]$
$= (4 x - 2 y + 3 z)^{2} = (4 x - 2 y + 3 z) (4 x - 2 y + 3 z)$

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Question 29
Factorise the following:

9 x^{2} + 4 y^{2} + 16 z^{2} + 12 x y - 16 y z - 24 x z

25 x^{2} + 16 y^{2} + 4 z^{2} - 40 x y + 16 y z - 20 x z

16 x^{2} + 4 y^{2} + 9 z^{2} - 16 x y - 12 y z + 24 x z

9 x^{2} + 4 y^{2} + 16 z^{2} - 12 x y - 16 y z + 24 x z

9 x^{2} + 4 y^{2} + 16 z^{2} + 12 x y + 16 y z + 24 x z

9 x^{2} + 4 y^{2} + 16 z^{2} + 12 x y - 16 y z - 24 x z

Factorize:

$16 x^{2} + 4 y^{2} + 9 z^{2} - 16 x y - 12 y z + 24 x z$

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