In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equationThe quadratic form f (x, y) = ax 2 + bxy + cy 2 is a solution of the above functional equation.

We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation 2f x y, z w /2 f x, z f x, w f y, z f y, w .

LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

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