In this paper, we introduce and solve of the radical quadratic and radical quartic functional equations: \begin{equation*} %\label{re2} f(\sqrt{ax^2+by^2})=af(x)+bf(y),\\ \end{equation*} \begin{equation*} %\label{re4} f(\sqrt{ax^2+by^2})+f(\sqrt{|ax^2-by^2|})=2a^2f(x)+2b^2f(y). \end{equation*} We also establish some stability results in $2$-normed spaces and then the stability by using subadditive and subquadratic functions in $p$-$2$-normed spaces for these functional equations.
