Quadratic regression models are strong that can be used to capture nonlinear effects of variables but their estimation is normally based on the conventional least-square techniques which may not be efficient to deal with complex data structures or noise. The study will suggest a superior model in estimating quadratic regression models using Artificial Neural Networks (ANNs) to increase the accuracy of prediction, and the strength of the model. The paper presents a mixed method that combines the theoretical characteristics of the quadratic regression with the adaptive learning of ANNs. The proposed model does not need explicit poly feature engineering to learn the desired nonlinear and linear interaction between predictors through training neural architectures as approximations of quadratic functions. Due to the large size of the experiments, synthetic data (through simulation of a known quadratic process) and real-world datasets were used to test performance in the presence of different levels of noise and data distributions. This study evaluates the proposed model using both synthetically generated data and real-world datasets obtained from verified institutional and laboratory sources within Iraq. Findings indicate that the ANN based quadratic estimator is much better than the classical regression methods in mean squared error, generalization and computational power. This contribution presents a new direction in the current regression modeling, a solution that can be scaled up and can connect both the statistical and machine learning paradigms of nonlinear data analysis.