Question
Factorise:
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
✓
Answer
We have:
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
$=(4 x)^2+(-2 y)^2+(3 z)^2+2(4 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(4 x)$
$=(4 x-2 y+3 z)^2\left[\text { using } a^2+b^2+c^2+2 a b+2 b c+2 c a=(a+b+c)^2\right]$
Hence, $16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z=(4 x-2 y+3 z)^2$
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
$=(4 x)^2+(-2 y)^2+(3 z)^2+2(4 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(4 x)$
$=(4 x-2 y+3 z)^2\left[\text { using } a^2+b^2+c^2+2 a b+2 b c+2 c a=(a+b+c)^2\right]$
Hence, $16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z=(4 x-2 y+3 z)^2$
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