This letter aims at complementing previous empirical work regarding a certain beamforming technique for blind speech extraction that uses a local quadratic approximation of a Kurtosis expression. It is shown here that the proposed method possesses a fixed-point property which means that it remains at an optimal solution once this solution has been reached. The proposed method's fixed-point property is valid for a range of source signals including Gaussian sources. This is an improvement over the FastICA method which diverges at the optimal points that correspond to a Gaussian source. In a real application, it cannot be assured that non-Gaussian mixtures are constantly observed; hence, the proposed method is a viable alternative in that case. The fixed-point property further implies that the approximative Kurtosis expression is identical to the true Kurtosis value at an optimal point which, in turn, means that the approximation error is zero. In addition, the convergence towards an optimal solution is always in the direction of a local minimum point even though the optimal solution that correspond to a super-Gaussian source is always a maximum solution which harmonizes with the concept of Kurtosis maximization.