How do you know what software was used? Or even if software was used. Wolfram Alpha, Sage and Geogebra are available to everyone for free, why do you call them exotic? You can use any language, even a graphing calculator to do this. But it still does not address the real question - "How many roots are there?"
The simplest idea is to use numerics, that means plot it first.
This immediately gives initial estimates for iteration.
You can explore a bit more with graphing to convince yourself that there are only two and this I felt would be adequate for this problem.
But, locating all the roots in non polynomial equations can be tricky. One way to prove that these are the only two would be to show that the curves continue decreasing and therefore never touch the x axis again.
Of course, whether or not this would be sufficient depends on where this problem originates. If it is a school or contest problem then there is almost without a doubt a clever idea that reduces the problem to an easy and clean solution. Therefore, I would continue to look for it.
But, if this problem originates from the real world we might have to be satisfied with the above suggestions.
Wait a second, there is someone in here right now that would take great delight in doing what she thought I could not. I will cease my efforts and let her succeed, only for the sake of chivalry of course.
If you had 5 dollars and your bills were 7 dollars do you not owe 2 dollars? Owing money is one practical use of negative numbers.
Did you read this?
It is pretty well explained out with nice diagrams. There you will see what I meant by counting left.