Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras \begin{equation} F~\xrightarrow{\Delta} ~F\times F~ \xleftarrow{f} ~F[X], \end{equation} $f(X)=(0,1)$. Geometrically this is $\mathbb{A}^1_F$ with $0$ and $1$ identified.

$A$ looks like the node $F[X,Y]/(X^2+X^3-Y^2)$. Is $A$ actually isomorphic to a quotient of $F[X,Y]$?