Exponential equations in free groups were studied initially by Lyndon and Schutzenberger and then by Comerford and Edmunds. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential equations have solution is decidable, in finitely generated free groups. In this paper the author shows that for finite systems of quadratic exponential equations decidability passes, under certain hypotheses, from the factor groups to free products and one-relator products.