= Solution
Take the <projective hypersurface>
$$
\boxed{S=V_+(X_0X_1-X_2^2)\subseteq\mathbb P^3_{\mathbb C}.}
$$
As a polynomial in $X_0$ over $\mathbb C[X_1,X_2,X_3]$, it is primitive because $X_1$ and $X_2^2$ are coprime. It is linear and irreducible over the fraction field, so <Gauss lemma for polynomials> proves irreducibility. Thus it is an irreducible surface.
Its gradient is $(X_1,X_0,-2X_2,0)$. The simultaneous zero set on $S$ is exactly
$$
\boxed{\operatorname{Sing}S=\{[0:0:0:1]\}.}
$$
The <Jacobian criterion> proves every other point is smooth. At that vertex the affine equation in the chart $X_3=1$ has no linear term, so the tangent space has dimension three whereas the surface has dimension two; the vertex is genuinely singular. This is a <projective quadric cone with one singular vertex>.
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