The functional equations $ f^2+g^2=1 $ and $ f^2+2αfg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. Utilizing Nevanlinna's theory for several complex variables, in this paper, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form $ af^2+2αfg+b g^2+2βf+2γg+C=0 $ in $ \mathbb{C}^2 $. Consequently, we obtain certain corollaries of the main results of this paper concerning binomial equations which generalize many results in [\textit{Rocky Mountain J. Math.} \textbf{51}(6) (2021), 2217-2235] in the sense of arbitrary coefficients.

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