Express $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 z x$ as a square of a trinomial.

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Solution

Using the identity: $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$ .

Given: $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 z x$ we can write it as follows:

$4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 z x = (2 x)^{2} + (3 y)^{2} + (- 4 z)^{2} + 2 \times 2 x \times 3 y + 2 \times 3 y \times - 4 z + 2 \times - 4 z \times 2 x$
$= (2 x + 3 y - 4 z)^{2}$

Hence, given expression $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y - 24 y z - 16 z x$ can be written as square of $(2 x + 3 y - 4 z)^{2} .$

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

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

Factorise:

(i)

(ii)

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)

(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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