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Thus, the number of solution pairs of the system is if the parameters and are chosen from the blue or green cells, to the right of the parabola, and otherwise.
You can restrict the cells for which you want the number of solutions.
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In fact, the system has exactly one solution, of multiplicity , on the parabola itself. This cannot be inferred from the previous data. However, you can add the equation for the discriminant variety to the system and make an indeterminate as well.
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In order to proceed, you must remove multiplicities by computing the radical.
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Notice that there is only one cell, and that the equations have two solutions, independent of the value of the parameter . The following command computes these solutions for :
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Since the values of are different for each of those two solutions, you can conclude that the original system has exactly one solution for parameter values and on the parabola specified by the equation for the discriminant variety.