Summary: Several researchers have recently derived formulae for economic order quantities (EOQs) with some variants without reference to the use of derivatives, neither for first-order necessary conditions nor for second-order sufficient conditions. In addition, this algebraic derivation immediately produces an individual formula for evaluating the minimum expected annual cost. The purpose of this paper is threefold. First, this study extends the previous result to the EOQ formula, taking into account the scenario where the quantity backordered and the quantity received are both uncertain. Second, the complete squares method can readily derive global optimal expressions from a non-convex objective function in an algebraic manner. Third, the explicit identification of some analytic cases can be obtained: it is not as easy to do this using decomposition by projection. A numerical example has been solved to illustrate the solution procedure. Finally, some special cases can be deduced from the EOQ model under study, and concluding remarks are drawn.