NSolve missing solutions

Question

I'm having trouble solving the transcendental equation. For some values ​​of bi, NSolve obtains 6 roots, however when changing the value, it obtains 5. Graphing the function clearly shows that the root that is missing exists and is in the interval between 10 and 11.

This is the code running all the roots correctly

bi=0.515625;
g[β_]:=(β*BesselJ[1, β])-(bi*BesselJ[0,β]);
roots=(β/.NSolve[{g[β]==0,0<=β<=18},β])
Plot[g[β], {β, 0, 18}, PlotRange -> {-5, 5},
 Epilog -> {Red, AbsolutePointSize[6], Point[{#, 0} & /@ roots]}]

And this is the same code with another bi value

bi=1.27188;
g[β_]:=(β*BesselJ[1, β])-(bi*BesselJ[0,β]);
roots=(β/.NSolve[{g[β]==0,0<=β<=18},β])
Plot[g[β], {β, 0, 18}, PlotRange -> {-5, 5},
 Epilog -> {Red, AbsolutePointSize[6], Point[{#, 0} & /@ roots]}]

Answer 1

Clear["Global`*"]

Functions should use explicit arguments for all variables.

g[bi_, β_] := (β*BesselJ[1, β]) - (bi*BesselJ[0, β]);

A function that uses a numeric technique (e.g., NSolve) should use SetDelayed rather than Set and have its arguments restricted to numeric values.

EDIT: Used Rationalize on argument and used Solve rather than NSolve

roots[bi_?NumericQ] := Module[{bir = Rationalize[bi, 0]},
  β /. Solve[{g[bir, β] == 0, 0 <= β <= 18}, β] // 
   Quiet]

roots[0.515625] // N

(* {0.953584, 3.96325, 7.08857, 10.224, 13.3623, 16.5019} *)

Plot[g[#, β], {β, 0, 18},
    PlotRange -> {-5, 5},
    PlotLabel -> StringForm["bi = ``", #],
    Epilog -> {Red, AbsolutePointSize[6],
      Point[{#, 0} & /@ roots[#]]},
    ImageSize -> 300] & /@
  {0.515625, 1.27188} // Column