Approximate solutions for a electron in uniaxially strained graphene have been determined. A first-order Taylor expansion for the Fermi velocities and the pseudo-vector potential, in the effective Hamiltonian describing uniaxially strained graphene, allows us to solve the corresponding differential equation of the eigenvalue problem. A finite number of bound states have been found and its spectrum is compared with the zero-order approximation derived in [1]. It turns out that the first-order approximation generated less energy levels than the zero-order approximation, revealing a suppression mechanism that deletes some energy levels.