The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$

Enter all the possible values, separated by commas.
Question:
The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$

Enter all the possible values, separated by commas.
Answer:
Let's think step by step.

user
0,\tfrac32, 2i, -2i
The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$

Enter all the possible values, separated by commas.
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