A node has two distinct tangent directions

The polynomial $f=y^2-x^3-x^2\in k[x,y]$ has the unique monomial expansion $y^2-x^3-x^2$ by [F2], with homogeneous parts $f_2=y^2-x^2$ of degree $2$ and $f_3=-x^3$ of degree $3$ by [F3], and it generates the principal ideal $I=(f)$ by [F12]; the quotient ring $A=k[x,y]\mathrm{/}(f)$ defines the closed subscheme $C=\mathrm{Spec}A$ of ${\mathbb{A}}_k^2$, and $k[x,y]=k[x][y]$ is a commutative polynomial ring by [F1].

The multiplicity of the equation $f$ at the origin is $2$: the coefficient of $y^2$ in $f$ is $1\mathrm{\ne }0$ by [F2] and [F14], so $f\mathrm{\ne }0$; every monomial of $f$ has total degree at least $2$, so $f$ has no constant term and $f(0)=0$; the translated expansion $f(t_1,t_2)=t_2^2-t_1^3-t_1^2$ is $f_2+f_3$ with $f_2=t_2^2-t_1^2\mathrm{\ne }0$ homogeneous of degree $2$ and $f_3=-t_1^3$ homogeneous of degree $3$, and there is no part in degree $0$ or $1$, so the least $j$ with $f_j\mathrm{\ne }0$ is $j=2$ and ${\mathrm{mult}}_0(f)=2$ by [F4].

The origin is a $k$-rational point of $C$: the evaluation map $\epsilon :k[x,y]\to k$ at $(0,0)$ has kernel $(x,y)$, which is maximal by [F9]; $f\in (x,y)$ because $f$ has no constant term, so [F10] factors $\epsilon$ through a $k$-algebra homomorphism $A\to k$, and [F11] exhibits the origin as a $k$-rational point $0=(0,0)$ of $C$.

The scheme-theoretic tangent cone: $f\in \mathfrak{q}=(x,y)$ and $I=(f)$ with $f\mathrm{\ne }0$, so [F5] applies with $P=k[x,y]$ and gives ${\mathrm{Cone}}_0(C)\cong \mathrm{Spec}(k[x,y]\mathrm{/}{\mathrm{in}}_{\mathfrak{q}}(I))$ with ${\mathrm{in}}_{\mathfrak{q}}(I)=(f_{\min })=(y^2-x^2)$ in the principal case; by [F6] this spectrum is the scheme-theoretic tangent cone, so ${\mathrm{Cone}}_0(C)\cong \mathrm{Spec}(k[x,y]\mathrm{/}(y^2-x^2))$, presented by its actual ideal with no reduction performed.

The tangent space is $k^2$: since $I=(f)$ is generated by the single equation $f=y^2-x^3-x^2$, the Jacobian matrix of the list $(f)$ at the origin is the $1\times 2$ matrix $(-3x^2-2x,2y)$ evaluated at $(0,0)$, namely $(0,0)$, by [F7]; its kernel is all of $k^2$, so [F8] gives a canonical $k$-linear isomorphism $T_{0}C\cong \ker (0:k^2\to k)=k^2$. Both entries are multiples of $x$ or of $y$ and vanish at the origin in every characteristic, so the computation does not use the nonvanishing of $2$ or $3$.

The two factors: in $k[x,y]$, $(y-x)(y+x)=y^2+yx-xy-x^2=y^2-x^2$ by distributivity and commutativity [F1]; if the two linear forms were proportional, then $y-x=c(y+x)$ for some $c\in k$, and comparing the coefficients of $y$ and of $x$ in the unique expansion [F2] gives $1=c$ and $-1=c$, hence $1=-1$ in $k$; that equality means $1+1=0$, contradicting $\mathrm{char}k\mathrm{\ne }2$ by [F13] together with $1\mathrm{\ne }0$ by [F14]. Hence $y-x$ and $y+x$ are non-proportional, and the cone $\mathrm{Spec}(k[x,y]\mathrm{/}((y-x)(y+x)))$ of step 3.2 presents the two distinct lines $y=x$ and $y=-x$ through the origin, each factor entering once in the degree-two leading form.

Conclusion: at the origin of the nodal plane cubic $C$ defined by $y^2=x^2+x^3$ the multiplicity of the equation is $2$ (step 2.1), the tangent space is $T_{0}C\cong k^2$ (step 4.1) and the scheme-theoretic tangent cone is $\mathrm{Spec}(k[x,y]\mathrm{/}(y^2-x^2))$, the two distinct lines $y=x$ and $y=-x$ (steps 3.2 and 4.2), so the singularity has two distinct tangent directions and its leading form is neither a power of one line nor an irreducible quadratic.

Boundary and scope dispositions: $C$ is nonempty because the origin is a $k$-rational point of it by step 3.1, and the cone is a spectrum over $k$ presenting the two lines of step 4.2, so no object here is empty; the zero cases are the vanishing Jacobian row $(0,0)$ of step 4.1 with kernel $k^2$, the zero constant and linear parts of $f$ in step 2.1, and the zero $xy$-coefficient in $y^2-x^2$ used up in the coefficient comparison of step 4.2; there is one equation, one Jacobian row, one cone equation and one occurrence of each linear factor (steps 1.1, 3.2, 4.1, 4.2); the degenerate case is exactly characteristic $2$, where $y^2-x^2=(y+x{)}^2$ and the two factors collapse to a doubled line, so the hypothesis $\mathrm{char}k\mathrm{\ne }2$ is used once, at step 4.2, to make the two tangent directions distinct, while multiplicity and tangent space are computed without it; the endpoints of the degree filtration are the least nonzero degree $2$ and the top degree $3$ of $f$ by step 2.1; no Axiom of Choice or dependent choice is used, since the equation, the Jacobian, the initial form and the coefficient comparison are all exhibited explicitly and every cited supplier is choice-free; and no biconditional is asserted or used, non-proportionality in step 4.2 being obtained by a one-directional coefficient comparison from a hypothetical proportionality, not by an equivalence.