Quadratic formula and isolating $f(x)$

Question

$f(x) - \frac{1}{f(x)} = g(x)$

I need to isolate $f(x)$. So far I got

$f(x)f(x) - g(x)f(x) - 1 = 0$

Now I've tried to use the quadratic root formula with no success.. What should I do?

Answer 1

As the comments indicated, replace $f(x)$ by $X$ and $g(x)$ by $a$. So we have to solve $$X-\frac{1}{X}=a$$ Multiplying both sides by $X$, we get $$X^2-1=aX$$ Using the quadratic formula, we get $$X=\frac{a\pm\sqrt{a^2+4}}{2}$$ So $$f(x)=\frac{g(x)\pm\sqrt{g(x)^2+4}}{2}$$