Asked by Bob
Describe all solutions to$$zw -3w - 2iw + 4iz = - 8 + 12i$$where $z$ and $w$ are complex numbers
All Answers 1
Answered by
GPT-5 mini
AI
Rewrite the equation as
z w - 3w - 2i w + 4i z + 8 - 12i = 0.
Group and factor:
(z - (3+2i))(w + 4i) = z w + 4i z - (3+2i)w - 4i(3+2i)
= (z w - 3w - 2i w + 4i z) + 8 - 12i,
so using the given equation the left side equals 0. Hence
(z - (3+2i))(w + 4i) = 0.
Therefore either z = 3 + 2i (with w arbitrary) or w = -4i (with z arbitrary). The pair (z,w) = (3+2i,-4i) lies in both families. These are all the solutions.
z w - 3w - 2i w + 4i z + 8 - 12i = 0.
Group and factor:
(z - (3+2i))(w + 4i) = z w + 4i z - (3+2i)w - 4i(3+2i)
= (z w - 3w - 2i w + 4i z) + 8 - 12i,
so using the given equation the left side equals 0. Hence
(z - (3+2i))(w + 4i) = 0.
Therefore either z = 3 + 2i (with w arbitrary) or w = -4i (with z arbitrary). The pair (z,w) = (3+2i,-4i) lies in both families. These are all the solutions.
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