A complete sentence is one that has a grammatical subject and a full verb, and that expresses a "complete idea." These are the minimal ingredients of a complete sentence. Anything else is a fragment.
All you have to do when identifying a fragment is ask whether it has a grammatical subject and at least one full verb (e.g. not just an -ing or -ed form alone). If one of these elements is missing, the utterance is a fragment.
Not all sentence fragments are inadvisable, but they should be avoided, except for special purposes such as emphasis, or in colloquial writing.
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