How to Find Turning Points of Particle Motion on a Smooth Cone?

[misterpickle]

Homework Statement

Consider the motion of particle moving on the inside surface of a smooth cone of half-angle α, subject to the gravitational force. Although this problem does not involve a central force, certain aspects of the motion are the same as for a central-force motion.

Show that the turning points of the motion can be found from the solution of a cubic
equation in r.

Homework Equations

After solving the Hamiltonian I have found the cubic equation.

$$mgr^{3}cot\alpha - Er^{2} + frac{l^{2}}{2m} =0$$

where $$l$$ is the generalized angular momentum and E is the energy of the system.

*** the third term should read (l^2)/(2m)

The Attempt at a Solution

I know that if I assume a E=const I can find 3 roots, 2 of which are real. However I have never taken a course in nonlinear physics and only know how to solve analytical cubic equations. I'm sure Maple is the easiest way to do this, but my Googling has turned up nothing but analytics. Can someone help me in solving this?

 

[The legend]

To find roots of any cubic equation you can solve using cardano's method
http://en.wikipedia.org/wiki/Cubic_function
examples:
http://www.trans4mind.com/personal_development/mathematics/polynomials/cardanoMethodExamples.htm