Hello I don t know much about geometrie but I want to remark
that the given quintic :

x^5 -5t*x^3 + 5t^2*x -2q = 0 is always solvable(!)

(thus for any value of b )

as a sum of two quintic roots. ( b = 2q ) ( t = 1 )

x = u + t/u will do.

x = (q + (q^2 -t^5 )^(1/2) )^(1/5) + (q - (q^2 -t^5 )^(1/2) )^(1/5)

Gives a perfect solution.

I think it is not possible to constuct with ruler and compass.

Greetings Ruben Hogenhout