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Write your equation as:I was unable to see the solutions in this parameter space.
A*exp(-r/t) = (b*t-c)/(1-b*t+c)
Choose the constants and Plot each side on the same graph.
Example: Choose A= -2, b=0.2, c=1, r=1 and plot from t=-20 to t=20, avoiding the singularities at t=0and t=10.
Three points of intersection appear, but beware of intersections at singularities.
Bob Sciamanda