Reduced Cubic

For what values of p does the equation

have 3 real roots? Extend for other values of q (q = 1 in the above equation). Summarize.

Take a look at the graph in the xp plane. This is a graph of the roots.


Consider what happens when p = -3.

The line crosses the graph in 3 places, each of which is a root. The first value of p which gives 3 roots is between -1.5 and -2.5 (I'm just eyeballing the values). When the horizontal line is tangent to the curve on the right, there will be two roots. When the x value is greater than that, there will be only one root. Everything less than this value (whatever it is) will give me 3 roots. Let's see if we can get more exact.

Here are several graphs for different q (still in the xp plane).

Let's look at the locus of these maximum points, the graph will be a parabola like this:

This parabola is the equation


This equation shows us all the p values on one graph, instead of having to graph each one with different q.

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