Explicit formulae for inhomogeneous binary quadratic congruences modulo prime powers

In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence $a{x}^2+bxy+c{y}^2+dx+ey\equiv n(\mathrm{mod}q),$ where $q$ is a prime power. We treat the cases where $q$ is an odd prime power and a power of 2 separately. By combining exponential sum methods, quadratic Gauss sums, and Hensel's lemma, we obtain exact formulae for the number of unrestricted solutions over ${\left({\mathbb{Z}}_q\right)}^2$ and the number of solutions with both variables restricted to units over ${\left({\mathbb{Z}}_q^{\ast }\right)}^2$. This paper simultaneously handles a mixed quadratic term and two separate linear terms, extending several previously studied diagonal, homogeneous, and symmetric special cases.