Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors.
Define an equivalence relation on the set of alline plane conics by: two conics are equivalent if tere is a polynomial change of variables that maps one conic to the other one. How do I describe the set of equivalence classes under this relation? I have no idea how to approach this.