 Solution

Use coefficient coordinates $[a:b:c]$ on $\mathbb{P}(V)$. A nonzero binary quadratic has a repeated linear factor precisely when its discriminant vanishes, soIts gradient is $(-4c,2b,-4a)$, which cannot vanish at a projective point. Therefore $C$ is a nonsingular projective conic. The parametrization $[\alpha :\beta ]\mapsto [{\alpha }^2:2\alpha \beta :{\beta }^2]$ identifies it as a degree-two Veronese map.