When do you use anjom dadan?
I know it means do but in what context. When would you use it as opposed to kardan etc.?

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I guess کردن is much more commonly used, while the other is used more formally.

Also, انجام دادن somehow implies to do and finish something, while کردن usually just implies doing it.

Another point is that کردن can be combined with other words to make many new verbs, such as: گریه کردن = to cry, تکرار کردن = to repeat, پیدا کردن = to find, مخلوط کردن = to mix, خرج کردن = to spend, فرار کردن = to run away, etc.

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I just remembered something, for the passive, we don't use Kardan. Instead we use anjam dadan. For example: 
این کار باید انجام بشه.

but this would be wrong: 
این کار باید کرده بشه.

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anjam dadan(not anjom dadan)is more formal

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وقتی میخوای رسمی تر باشی
when you wanna sound more formal