This paper proposes the use of two task-aware warping factors in mask-based speech enhancement (SE). One controls the balance between speech-maintenance and noise-removal in training phases, while the other controls SE power applied to specific downstream tasks in testing phases. Our intention is to alleviate the problem that SE systems trained to improve speech quality often fail to improve other downstream tasks, such as automatic speaker verification (ASV) and automatic speech recognition (ASR), because they do not share the same objects. It is easy to apply the proposed dual-warping factors approach to any mask-based SE method, and it allows a single SE system to handle multiple tasks without task-dependent training. The effectiveness of our proposed approach has been confirmed on the SITW dataset for ASV evaluation and the LibriSpeech dataset for ASR and speech quality evaluations of 0-20dB. We show that different warping values are necessary for a single SE to achieve optimal performance w.r.t. the three tasks. With the use of task-dependent warping factors, speech quality was improved by an 84.7% PESQ increase, ASV had a 22.4% EER reduction, and ASR had a 52.2% WER reduction, on 0dB speech. The effectiveness of the task-dependent warping factors were also cross-validated on VoxCeleb-1 test set for ASV and LibriSpeech dev-clean set for ASV and quality evaluations. The proposed method is highly effective and easy to apply in practice.