A class of third order methods for solving nonlinear equations, containing in particular the Euler–Chebyshev method, is discretized in such a way that the second derivatives $F''(x^k )$ of the function F are approximated by bilinear operators $B_k $. The approximations $B_k $ require only one, at most two, or exactly two Jacobians of F per step, so that the amount of work is not higher, or only slightly higher, than that of Newton’s method. Moreover, the operators $B_k $ are consistent; i.e., in particular, there holds $\lim (B_k - F''(x^k )) = 0$, and therefore the methods described converge superquadratically. Specific variants converge with order $\tau = \tau (n) > 2$ which is the positive zero of $\tau ^{n + 1} = 2\tau ^n + 1$.