Question

Factorise: 

(i) 4x ² + 9y ² + 16z ² + 12xy – 24yz – 16xz 

(ii) 2x ² + y ² + 8z ² – 2√2 xy + 4√2 yz – 8xz

Answer 1

Factorisation of the Given Expressions 

1. \( 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz \)

Step 1: Group terms cleverly to try completing squares:

\( 4x^2 + 12xy - 16xz + 9y^2 - 24yz + 16z^2 \)

Step 2: Factor coefficients where possible:

\( (2x + 3y - 4z)^2 \)

Answer: \( 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz = (2x + 3y - 4z)^2 \)

2. \( 2x^2 + y^2 + 8z^2 - 2\sqrt{2} xy + 4\sqrt{2} yz - 8xz \)

Step 1: Try to express as a square of a trinomial: Observe the coefficients and signs carefully.

Step 2: Factor terms and check for square: \( (\sqrt{2}x - y + 2z)^2 \)

Answer: \( 2x^2 + y^2 + 8z^2 - 2\sqrt{2} xy + 4\sqrt{2} yz - 8xz = (\sqrt{2}x - y + 2z)^2 \)