We consider the following quadratic and quartic functional equations \begin{align*} f\left(\sqrt{xx^*+yy^*}\hspace{.05cm}\right)=f(x)+f(y), \end{align*} \begin{align*} f\left(\sqrt[4]{(xx^*+yy^*)(xx^*+yy^*)^*}\right)+f\left(\sqrt[4]{(xx^*-yy^*)(xx^*-yy^*)^*}\right)=2f(x)+2f(y) \end{align*} in $C^*$-algebras. We also prove the stability of these functional equations in $\beta$-normed spaces by using the Banach fixed point theorem and a direct method.
