Region of Motion: Solving U(x) = ax^4 + bx^3 + cx + d with E=3

[tomwilliam]

Homework Statement

Find the region of motion of a particle in a system following a potential energy function of:
U(x)=ax⁴+bx³+cx + d
and total energy of E=3
I know the values of a, b, c and d.

Homework Equations

E_total=U(x)+E_kin

The Attempt at a Solution

I know that I can the turning points are where speed is zero, and therefore kinetic energy is zero, so:
E_total-U(x)=0 represents the boundary values of x.
So
3=ax⁴+bx³+cx + d
I've graphed the functions and can see the rough value of x which corresponds to the region of motion, but I need to provide an exact answer. I am not expected to solve quartic equations, so there must be a way of simplifying the potential energy function. I could differentiate it, but that's not really the solution I'm after. I could extract the factor of x (passing d over to the other side) but I don't know if I can remove the case where x=0.
Any help much appreciated
Thanks

Homework Statement

Homework Equations

The Attempt at a Solution

 

[lanedance]

unfortunately i think you have to solve the quartic 3=ax4+bx3+cx + d

what are the values of abcd? that might simplify things

 

[tomwilliam]

Thanks for your answer.
I was a bit wary about posting exact values, as this is a coursework question and I don't want to get in trouble with my course tutors. Can you think of any situations where a quartic equation like this could be simplified using some physics assumptions? (I should state that there is no x term, it was a typo, it should be cx²)
I know of some examples where you can set the equation to 0, then take a factor of x outside brackets and then solve the equation inside the brackets, but I've only seen it for cubic equations, and it doesn't seem to work for this equation as the value isn't 0.

Basically, I'm not at a level where I can solve quartic equations, and my tutors have already indicated that it's not necessary, so I guess I'm missing some basic assumption about the system.

Any help greatly appreciated, otherwise I'll just graph the function and read the values off that, but it won't be the algebraic solution they were after.
Thanks

 

[lanedance]

it would simplify a heap if d = 3...

but without seeing it i don't have any general tips besides what you've already mentioned

 

[tomwilliam]

Ok, well thanks for your help anyway.

I can make the equation

0 = ax⁴+bx³+cx²+e

but unfortunately a, b, c and e are all non-zero.
I'll try playing around with them and see if they factorise.
Thanks again