It is shown that the eigenvalue problem $u'' + Bu(\lambda ^2 + \lambda p)u$; $u(0) = u(1) = 0$ (where p is a positive function and B an arbitrary bounded operator on $L^2 [0,1]$ possesses in general two different sets of eigenfunctions, each of which is an unconditional basis for $L^2$ and other spaces. The method of proof, which is applicable to more general problems, uses the ordinary quadratic formula to find a factored operator-polynomial which is “close” to the original problem; perturbation techniques are then applied to derive the desired spectral information.