Quadratic and Cubic equations
Consider the graph of
Graph this relation in the xb plane.
To graph in the xb plane we must first write the relation as b = É.
We get b = -x – (1/x)
When we choose b=5 the graph has two negative real roots. For b greater than 2 the graph gives us two negative real roots. For b less than -2 the graph gives us two positive real roots and at b = 2 and b = -2 we get one negative real root and one positive real root respectively. For b between -2 and 2 we have no real roots.
The graph in the xb plane has two asymptotes. One is x = 0 and the other is b = -x.
Let us see how the curve draws out when c = -1.
The graph when c = -1 does not intersect the original graph. It has the same asymptotes as the original curve. This curve has two real roots throughout.
Lets try other values for c.
c = -4, c = 4, c = -1/2, c = ½
When c = 4 (green curve) we get a curve that has no real roots between -4 and 4 and for c = -4 (light blue curve) our curve is stretched out more and passes through 2 and -2 and has two real roots throughout.
For c = ½ (yellow curve) we get a curve that is closer to the origin than the original. It has no real roots between around -1.5 to 1.5. For c = -1/2 (gray curve) we get a curve closer to the origin also.