What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.

Usual completion of square breaks. I know For a method (different from checking all elements) for finite fields but field there is Chen Formula that roughly looks like $X=\sum_{m} c^{4^m}$. I need it for any am more interested in the local field $F((z))$ or actually an arbitrary field of characteristic 2.

2 added "in characteristic 2" to the title

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# How to solve a quadratic equation?

What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2?

Usual completion of square breaks. I know a method (different from checking all elements) for finite fields but I need it for any field of characteristic 2.