An equivalent lens is a lens which has the same total power of refraction and the same paraxial imaging characteristics for the marginal rays as another lens, but has a different axial thickness. In this study, an analytic lens conversion from a thick lens to its equivalent lens is investigated, then it is shown that the equivalent lens is a solution of a quadratic equation. Every thick lens corresponds to one of two real roots of this quadratic equation. Therefore, except in the case of a unique solution, the equation has a conjugate solution, the other of the two roots. The conjugate solution has the same axial thickness, power, and paraxial imaging characteristics, but it has different shape and aberration characteristics. The characteristics of an equivalent lens and its conjugate solution are examined by using a sample lens.