TITLE: Finding double root of $x^5-x+\alpha$
QUESTION [7 upvotes]: Given the polynomial
$$x^5-x+\alpha$$
Find a value of $\alpha>0$ for which the above polynomial has a double root.
Here's an animated plot of the roots as you change $\alpha$ from $0$ to $1$ I'm looking for $\alpha$ when the 2 points in the plot meet.

Also, this is not homework

REPLY [2 votes]: Somebody posted a hint answer which made solving it easy, but they quickly deleted it. Here's a solution using that answer.
When the discriminant of a polynomial is zero, atleast two roots coincide.
The discriminant of $x^5-x+\alpha$ is $3125\alpha^4-256$ solving for $\alpha$
$$3125\alpha^4-256=0$$
$$3125\alpha^4=256$$
$$\alpha^4=\frac{256}{3125}$$
$$\alpha=\frac{4}{\sqrt[4]{3125}}$$
$$\alpha=\frac{4}{5\sqrt[4]{5}}$$