Factorise: (i) `4x^2+9y^2+16 z^2+12 x y-24 y z-16 x z` (ii) `2x^2+y^2+8z^2-2sqrt(2)x y+4sqrt(2)y z-8x z`

Explanation

(i)` 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz`
This can be re-written as:
`(2x)^2 +(3y)^2 +(-4z)^2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x) `
This is of the form `a^2+ b^2+ c^2+ 2ab + 2bc + 2ca = (a + b + c)^2`
Here `a = 2x, b = 3y, c = -4z`

(ii) `2x^2+ y^2+ 8z^2- 2sqrt(2)xy + 4sqrt(2)yz - 8xz`
We know that, `2 = (sqrt(2))2, sqrt(8) = 2sqrt(2)`
Thus, the given expression can be re-written as:
`(-sqrt(2)x)^2 + (y)^2 + (2sqrt(2)z )^2 + 2(-sqrt(2)x)(y) + 2(y)(2sqrt(2)z) + 2(2sqrt(2)z)(-sqrt(2)x)`
This is of the form `a^2+ b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2`
Here `a = -sqrt(2)x, b = y, c = 2sqrt(2)z`.