jmc

Any solution to this equation must make the numerator of the left-hand side zero, while keeping the denominator non-zero. The numerator is zero when $x$ is one of the numbers $1,2,3,\dots ,100.$ However, for any value in this list that is a perfect square, the denominator will also be zero, so that value of $x$ will not be a root. Therefore, we want to find the number of integers in the list $1,2,\dots ,100$ which are not perfect squares. The perfect squares in the list are $1^2,2^2,\dots ,10^2,$ so there are $10$ perfect squares, and $$100-10=\boxed{90}$$integers which are not perfect squares.