In this work we propose a class of quasi-Newton methods to minimize a twice differentiable function with Lipschitz continuous Hessian. These methods are based on the quadratic regularization of Newton's method, with algebraic explicit rules for computing the regularizing parameter. The convergence properties of this class of methods are analysed. We show that if the sequence generated by the algorithm converges then its limit point is stationary. We also establish local quadratic convergence in a neighborhood of a stationary point with positive definite Hessian. Encouraging preliminary numerical experiments are presented.