Theorem: If p(x) and q(x) are polynomials and q(x0) ≠ 0, then the power series representation of p/q has radius of convergence R, where R is the distance, in the complex plane, from x0 to the nearest root of q.
We can take our q(x) as
Since q(x0) = 5 (not zero), we can use the theorem! Let’s use the quadratic equation to find the roots, z, of q(x)
Now we can use the Pythagorean theorem to find the distance in the complex plane from the center at x0
Note: the a, b variables above are the coefficients of the complex number z, separate from the a, b, c coefficients of q(x) used in the quadratic formula.