Heterogeneous (HE) multicore platforms, such as ARM big.LITTLE, are widely used to execute embedded applications under multiple and contradictory constraints, such as energy consumption and real-time (RT) execution. To fulfill these constraints and optimize system performance, application tasks should be efficiently mapped on multicore platforms. Embedded applications are usually tolerant to approximated results but acceptable quality of service (QoS). Modeling-embedded applications by using the elastic task model, namely, imprecise computation (IC) task model, can balance system QoS, energy consumption, and RT performance during task deployment. However, state-of-the-art approaches seldom consider the problem of IC task deployment on HE multicore platforms. They typically neglect task migration, which can improve the solutions due to its flexibility during the task deployment process. This article proposes a novel QoS-aware task deployment method to maximize system QoS under energy and RT constraints, where the frequency assignment (FA), task allocation (TA), scheduling, and migration are optimized simultaneously. The task deployment problem is formulated as mixed-integer nonlinear programming. Then, it is linearized to mixed-integer linear programming to find the optimal (OPT) solution. Furthermore, based on the problem structure and problem decomposition, we propose a novel heuristic (HEU) with low computational complexity. The subproblems regarding FA, TA, scheduling, and adjustment are considered and solved in sequence. Finally, the simulation results show that the proposed task deployment method improves the system QoS by 31.2% on average (up to 112.8%) compared to the state-of-the-art methods and the designed HEU achieves about 53.9% (on average) performance of the OPT solution with a negligible computing time.