Show that the addition of a term ax to the oscillator potential

V(x) = ax +21 m²x²,

results in a parabolic potential - the same parabola, translated horizontally in x and shifted down-wards in energy. Write down the new quadratic form (i.e. write V (x) =21 m²(x x₀) V₀ for some constants x₀ and V₀) and sketch the potential. Write down the energy eigenvalues of this new potential. (An electric field E in the x-direction acting on a particle of charge q would add a potential energy qEx to the oscillator potential, for example.)