Question 5
Factorise:
$(i) 4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z (i i) 2 x^{2} + y^{2} + 8 z^{2} - 2 \sqrt{2} x y + 4 \sqrt{2} y z - 8 x z$

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Solution

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

$(i i) 2 x^{2} + y^{2} + 8 z^{2} - 2 \sqrt{2} x y + 4 \sqrt{2} y z - 8 x z U s i n g t h e i d e n t i t y (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a = (\sqrt{2} x)^{2} + (- y)^{2} + (- 2 \sqrt{2} z)^{2} + 2 (\sqrt{2} x) (- y) + 2 (- y) (- 2 \sqrt{2} z) + 2 (\sqrt{2} x) (- 2 \sqrt{2} z) = (\sqrt{2} x - y - 2 \sqrt{2} z)^{2} = (\sqrt{2} x - y - 2 \sqrt{2} z) (\sqrt{2} x - y - 2 \sqrt{2} z)$

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Similar questions

(i) 4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 x z (i i) 2 x^{2} + y^{2} + 8 z^{2} - 2 \sqrt{2} x y + 4 \sqrt{2} y z - 8 x z

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

2 x^{2} + y^{z} + 8 z^{2} - 2 \sqrt{2} x y + 4 \sqrt{2} y z - 8 x z

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

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

= (2 x + 3 y - 4 z) (2 x + 3 y - 4 z)

2x squared + y squared + 8z squared - 2 root 2xy + 4 root 2yz - 8xz

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