$f(x) = 1 - 6/x + 8a/x^{3/2} - 3a^2/x^2$ where $-1<a<1$
How do you show that $f(x)$ has a root at $x>1+(1-a^2)^{1/2}$ for all $a$?
It can be shown that $f(x)$ has a local minimum at $x=a^2$ at which $f<0$ and that $f(x)$ approaches $1$ for large $x$, so the root must be located at $x>a^2$.