The Dixon technique is a well-established method in magnetic resonance imaging for obtaining separate images of water and fat. Here we present a generalized solution to the two-point Dixon problem with a geometric interpretation, allowing for flexible echo times and a multi-peak fat model. By simulation and experiment, we have analyzed the dependence on the echo times of the error in the water, fat, and relative background phasor values due to both signal noise and T2* decay. Furthermore, we have demonstrated that broken symmetry due to the multi-peak nature of fat enables direct water and fat determination without phase correction, and we have quantified the reliability of this technique as a function of the echo times. The results may provide valuable guidance for selecting scan parameters to balance the objectives of optimizing fat-water identification, minimizing error in the pixel values, and minimizing total scan time.