From the Work board, open the Task tile by selecting the Task summary in the Work items list. This will activate a Details Tile on the right-hand side of your screen.

From the Action Bar across the bottom, the File icon will allow you to update the Status of the Task.

Select Completed and Save.

If the Task contains Subtasks, you don't have to have all Subtasks checked to Complete the Task. Likewise, checking all the Tick Boxes alone will not Complete a Task.
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You will be shown the number of outstanding subtasks and be asked if you're sure you want to mark the task as complete. Click Yes or No.

The Completed Task will now appear on your Completed Items board.