 Hessian criterion for flexes of a plane cubic

For a smooth homogeneous cubic in characteristic zero, its plane cubic flexes are precisely its intersections with the determinant of its Hessian matrix. Choose coordinates with a given smooth point $(0:0:1)$ and its tangent $y=0$. Up to a nonzero coefficient, the cubic starts as $y{z}^2+a{x}^{2}z+bxyz+c{y}^{2}z$ plus cubic terms in $x,y$. Its tangent contact has order three exactly when $a=0$, and the Hessian determinant at the point is a nonzero constant times $a$. This proves the criterion.