Question
Factorise:$16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$
✓
Answer
We have:
$16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$
$= (4x)^2 + (-2y)^2 + (3z)^2 + 2(4x)(-2y) + 2(-2y)(3z) + 2(3z)(4x)$
$= (4x - 2y + 3z)^2$ [using $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2]$
Hence, $16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz = (4x - 2y + 3z)^2$
$16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$
$= (4x)^2 + (-2y)^2 + (3z)^2 + 2(4x)(-2y) + 2(-2y)(3z) + 2(3z)(4x)$
$= (4x - 2y + 3z)^2$ [using $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2]$
Hence, $16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz = (4x - 2y + 3z)^2$
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