Quadratic and Cubic
equations

Rozina Essani

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.