Let If then the positive term massively outweighs the other two terms, so that For all three terms are positive, so again
At x=2009 the function is increasing. It looks very much as though it then has a single maximum as x increases, after which it decreases monotonically to thereby crossing the x-axis just once and giving rise to the single real root of the function.
I am sure that there must be a more convincing way of attacking this problem. It should presumably take advantage of the fact that but I cannot see any way to make use of that fact.