$R=k[X] , f \in R$ . Let $A=k[X,Y]/(Y^2-f)$ . If $f$ has no square factor , then $A$ is a normal domain

Question

Let $k$ be a field , $R=k[X] , f \in R$ . Let $A=k[X,Y]/(Y^2-f)$ . I know $A$ is an integral domain iff $f$ is not a perfect square and hence forth assume $A$ is a domain. If $f$ has a square factor, then I can show that $A$ is not a normal domain. My question is : If $f$ has no square factor, then how to show that $A$ is a normal domain ?