an arbitrary degree 2 polynomial over an algebraically closed field has the form $(x-ay)(x-by)+cx+dy+e$ (up to scaling, which doesn't change the locus). if $a=b$, we substitute $x\mapsto x+ay$ and obtain $x^2+cx+d'y+e$. now, $d'\neq0$ as otherwise the polynomial is irreducible (which is preserved under affine linear automorphisms), so we can substitute $y\mapsto-\frac{1}{d'}(cx+y+e)$ and obtain $x^2-y$.
if $a\neq b\neq0$, i can substitute $x\mapsto x-\frac{a}{b}y$ and $y\mapsto\frac{1}{b}(x-y)$ to obtain $(1-\frac{a}{b})^2xy+c'x+d'y+e$. the coefficient is non-zero, so I can remove it by res…