How exactly do I solve arcsinx + x/sqrt[1-x] = 0???
You don't- not in terms of elementary functions, at least. You could, of course get an approximate solution by numerical methods. For example, you could solve this by graphing f(x)= arcsin(x)+ x/sqrt(1- x) on a graphing calculator and then "zooming" in on a point where the graph crossed the x-axis.