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Both "expeditious" and "expedited" can describe something done quickly, but they are used slightly differently in sentences:

1. Expeditious:
   - This is an adjective describing the manner or speed of the action itself.
   - Example: "The issue was resolved in an expeditious manner."
   - Here, "expeditious manner" means the manner was quick and efficient.

2. Expedited:
   - This is the past tense of the verb "expedite," meaning to speed up the process.
   - Example: "The process was expedited to meet the deadline."
   - Here, "expedited" describes the action of speeding up the process.

So, you should say:
- "The task was completed in an expeditious manner."
- "The task was expedited to ensure timely completion."

Choose "expeditious" if you want to describe how the task was done and "expedited" if you want to describe the action of speeding it up.