I don't think you are quite right: the function x^2 = 0 has only two solutions by Fundamental Theorem of Algebra, although in this case are the same; while sin(x) = 0 has indeed infinite solutions, as you stated, located at pi * k for any choice of an integer k. However, sin(x) is bounded and once x^2 is large enough, that is for |x| > 1 we can't have any more solutions to sin(x) + x^2 = 0 ! Indeed, as I said since |sin(x)| < 1 even in the best case, for |x| > 1 we would have x^2 - 1 > 0. Thus the only solutions are restricted in the domain [-1,1] and, as I said, are approximately x = -0.877 and x = 0. 

Finally, as you mentioned there's no general formula to find all roots of a continuous function, but methods using fixed point theorems could be used to determine whether a solution exists in a given interval. 

If you have any more questions on the matter, please let me know!
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence