This thesis will examine the theory of binary quadratic forms. Specifically, it will develop a series of algorithms which will be able to solve the representation problem for binary quadratic forms of negative discriminant. First, we must introduce several definitions and prove several results involving binary quadratic forms. Then we turn our focus to constructing forms of a given discriminant and leading coefficient. Finally, we use the reduction of binary quadratic forms in order to determine solutions of the representation problem. Based upon these results, we are able to provide insight in a future direction of how to generalize these algorithms to all binary quadratic forms.
