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In a recent paper, L. J. Mordell gave necessary and sufficient conditions for the equation      a x 2 + b y 2 = c a{x^2} + b{y^2} = c       to have algebraic integer solutions in the quadratic field      Q ( √  ( −  n ) ) Q(\surd ( - n))      . In this paper we drop the requirement that the solutions be algebraic integers. In particular, we prove that      a x 2 + b y 2 = c a{x^2} + b{y^2} = c       has solutions in      Q ( √  ( −  n ) ) Q(\surd ( - n))       if and only if the quadratic form      a b t 2 −  b c u 2 −  a c v 2 −  n w 2 ab{t^2} - bc{u^2} - ac{v^2} - n{w^2}       represents 0 over  Q .

The solvability of the equation 𝑎𝑥²+𝑏𝑦²=𝑐 in quadratic fields