On roots of quadratic congruences

The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for special Weyl linear forms. Duke, Friedlander and Iwaniec discovered strong estimates for these Weyl linear forms when the quadratic polynomial has negative discriminant. Tóth proved analogous but weaker bounds when the quadratic polynomial has positive discriminant. We establish strong estimates for these Weyl linear forms for quadratics of positive discriminants. As an application of our bounds, we derive a quantitative uniform distribution of modular square roots with integer moduli in an arithmetic progression.