The two-dimensional quadratic differential system (QDS) \[ \begin{gathered} \dot x = a_1 x^2 + b_1 xy + c_1 y^2 , \hfill \\ \dot y = a_2 x^2 + b_2 xy + c_2 y^2 \hfill \\ \end{gathered} \] where $( \cdot ) = {d} / {dt}$ and the coefficients are real constants is considered. Lyagina presented sixteen geometric equivalence classes for the integral curves of the associated scalar equation ${{dy} / {dx}} = {{\dot y} / {\dot x}} $. The application of this classification scheme depends upon making affine transformations so that the linear integral curves through the origin of the resulting equation will lie either along the y axis, the x axis, or on $y = x$. This amounts to transforming the given equation so that certain coefficients are zero. Lyagina then derived conditions on the remaining nonzero coefficients that yield the geometric equivalence classes for the integral curves, exhausting all of the possibilities. This paper classifies the trajectories of a given QDS without first making such an affine transf...