How to solve a simple equation with explanation

Question

I'm currently struggling to solve a simple equation because I'm rusty on my algebra skills. I made another post the other day which helped me greatly. That equation that @StubbornAtom helped me with worked but I'm afraid I misread their advice and am coming back to it now. In this equation I need to solve for $\alpha$ but am so rusty with cubic equations I can't figure it out. Can you please help and provide a step by step? In grad school I'll be sure to take refresher algebra courses. Thanks!

$$\sigma^2(4) = \frac{Mode^2(\alpha + 1)^2}{(\alpha - 1)^2(\alpha - 2)}$$

Answer 1

If you clear the fraction you have a fourth degree polynomial equation in $\alpha$. There is a formula for that which you can find if you search for quartic equation. It is complex enough that most people either feed it to Alpha or use numerical techniques to find an approximate answer. In the post you link the $\alpha-2$ term is not squared. That gives a cubic instead of a quartic. The algebraic solution is not as complicated, but still enough of a mess that many people use numerics.

Answer 2

Not an answer, just an insight on the number and positions of the real roots. This is a plot of

$$\frac{(\alpha+1)^2}{(\alpha-1)^2(\alpha-2)}.$$

The intersection with an horizontal give you from one to three solutions.

Caution: there are two additional extrema in the negatives (solve $9+5\alpha-5\alpha^2-\alpha^3=0$).