Quartic function of a non-ideal spring

[Semiclassical]

Homework Statement

Solve for y: $$\frac{\lambda}{4} y^4+\frac{k}{2} y^2+mgy-E=0$$

Relevant Equations

I do not know of any relevant equations

I'm stuck in a part of my problem where I need to find the roots of this function which represent turning points for a non-ideal spring.

 

[haruspex]

https://en.m.wikipedia.org/wiki/Quartic_equation