Not sure eliminated is a good word here, or that these concepts are only introduced to be it
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But those two (of the three) complex roots aren't useful to you unless you can eliminate them at some point. If you've introduced complex numbers as a means of describing those two roots, when you couldn't represent them before, that wasn't helpful if you can't get answers in...
...the system in which your original problem was defined. It might as well be the case that those two roots don't exist.
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Some problems have no solution in a space: you extend it, derive a solution in this “richer space” and prove its valid in the other (some approaches to measure separatrix splitting for real-valued dynamical systems involve this, for example: this is ripe in dynamical systems)
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Another example would be analysing collisions: they involve singularities that can be removed artificially by adding new re-scaled variables. Likewise with studying any problem close to collapse/smallness (perturbative approaches)
End of conversation
New conversation -
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"complex roots aren't useful to you unless you can eliminate them at some point" is just your (inexplicable to me) preference
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