How can I find the roots of x^2+px+q=0 by using contractive mapping theorem?

Question

Basically, Contractive mapping theorem guarantees that if F is contractive on some interval, then there exists a unique fixed point.

However, $x^2+px+q=0$ has two different solutions as long as $p^2>4q$

Either two roots are negative, or one is positive and the other is negative.

In this case, how can I generalize the intervals for each root ?

I considered this function $f(x)=-\frac{q}{p+x}$

And $f'(x)= \frac{q}{(p+x)^2} $

This is exactly where I am stuck