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
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: \( 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz = (2x + 3y - 4z)^2 \)
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: \( 2x^2 + y^2 + 8z^2 - 2\sqrt{2} xy + 4\sqrt{2} yz - 8xz = (\sqrt{2}x - y + 2z)^2 \)