M2: Reciprocal Roots

This is a user suggested problem. The TopsOJ staff thanks and gives full credit to Maximilian113 for their contribution.

Let q(x) be a monic quadratic. Suppose q(x) has roots r_1, r_2 satisfying the following properties:

1. r_1 =\frac{1}{r_2}
2. r_1, r_2 are of the form a \pm b \sqrt{c}, where a, b, c are positive integers, c is not divisible by the square of any prime, and a = b + c.

Find the sum of all possible values of q(-2).