Solution
= 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, so
$$
\boxed{C=\{[a:b:c]:b^2-4ac=0\}}.
$$
Its <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>.