Integers of the Form $a{x}^2+bxy+c{y}^2$

We provide a characterization of integers represented by the positive definite binary quadratic form $a{x}^2+bxy+c{y}^2$. In order to prove the main theorem, we define the “relative conductor” of two orders in an imaginary quadratic field. Then we provide a characterization of decomposition of proper ideals of orders in imaginary quadratic fields. Moreover we present some interesting examples of the main theorem.