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Let S be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution f : ℝ2 → S of the following parametric functional equation f ( x 1 + x 2 + α y 1 y 2 , x 1 y 2 + x 2 y 1 + β y 1 y 2 ) = f ( x 1 , y 1 ) f ( x 2 , y 2 ) , f\left( {{x_1} + {x_2} + \alpha {y_1}{y_2},{x_1}{y_2} + {x_2}{y_1} + \beta {y_1}{y_2}} \right) = f\left( {{x_1},{y_1}} \right)f\left( {{x_2},{y_2}} \right), for all (x 1, y 1), (x 2, y 2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.

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