We develop two new training algorithms for feed forward supervised neural networks based on quadratic optimization methods. Specifically, we approximate the error function by a quadratic convex function. In the first algorithm, the new error function is optimized by an affine scaling method, which replaces the steepest descent method in the back propagation algorithm. In the second algorithm, the steepest descent method is replaced by a trust region technique. Comparative numerical simulations for medical diagnosis problems show significant reductions in learning time with respect to the back propagation algorithm.