Answer
$$ \frac{1+ \sqrt5}{2}$$
Work Step by Step
We simplify:
$$x(x-1)=1$$
$$x^2-x-1=0$$
We use the quadratic formula:
$$x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
$$x= \frac{1 \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)}$$
This gives:
$$ \frac{1+ \sqrt5}{2}$$
We only consider this answer since the ratio must be positive.