Originally posted by mlprior
I have a TI-86 but don't have the manual, maybe I can find the directions online.
Whilst you're playing around with graphing calculators and such; assuming you don't have money to burn grab yourself a copy of maxima 5.24 (poor man's mathematica but it's actually pretty good, and is actually the distant ancestor of the likes of maple and mathematica)
Also grab yourself a copy of Scilab (free clone of MATLAB - not quite as efficient but again, it's pretty good)
Then use these instead to do your plotting and root finding (and whatever else) - save the batteries on your TI :]
As for your OP, another pencil & paper approach might to perturb a trial solution x=1 with a small value a; i.e. evaluate 10(1+a)^9 -9(1+a)^10 = 3/4 and when you expand it you'll have a different degree 10 polynomial where you know already that the higher order terms are neglible (since |a| is small) and so you can throw some of these away - say up the the 4th power of a perhaps, and play around with that instead.