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The other day I asked a question in here about deriving the equations $$u^2\left(\left(1-s_1\right)+3u+3u^2+u^3\right) =\alpha\left(s_0+2s_0u+\left(1+s_0-s_1\right)u^2+2u^3+u^4\right),$$

where $s_0=sgn(u)$ and $s_1=sgn(u+1)$.

Now I seek approximate solutions to the quintic equations, when $0<\alpha<<1$

I know pretty much nothing about solving equations/polynomials numerically. I would appreciate, if you guys in here could give me a link and some brief explanations, or possibly solve the whole thing if it is not too complicated.

The results are written on page 4 in these notes: http://www.physics.montana.edu/faculty/cornish/lagrange.pdf

I would of course like to know, how the results are derived, thank you.

EDIT: It should be possible to reduce the equation and somehow do a taylor expansion in $\alpha$ to come to an answer. Also see Rewriting to quintic equation, which might tell something about how $R$ comes into the answer.

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