Your method of attack seems very interesting but is unnecessary.
Note that for x=0
which is one of the intersections.
Now for x>0 we have
This intersects the real axis when the imaginary component is zero. This happens when
Now solve for x.
Let y(x) be a real valued function defined on the interval by means of the equations:
show that the equation representing an arc that intersects the real axis at the point and
I'm not sure if I'm picking the right values for this, but I would think that it would something along the lines of:
this is the part that I'm not sure of, but continuing would give me:
putting the above equation into Maple gave me:
for some reason this doesn't look correct.