...negative quantity of any material thing, but we can eliminate negative numbers by combining them with other numbers (positive or negative) to get back to a positive number which we can understand. Complex numbers are similar.
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Sorry, if the solution to a cubic is complex, its complex. You have a solution. You just seem to think its neater not to be in extended system somehow
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...
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...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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