Is this grammatically correct?
The teacher has sent several students home because they cheated on the quiz.
Or does it have to be like this?
The teacher has sent several students home because they have cheated on the quiz.
Is this grammatically correct?
The teacher has sent several students home because they cheated on the quiz.
Or does it have to be like this?
The teacher has sent several students home because they have cheated on the quiz.
The teacher has sent several students home because they cheated on the quiz.
Some students cheated on the quiz. This was later discovered (by the school/teacher) and consequently they were suspended (implied) and sent home. We do not know when the students cheated, but presumably it was not on the same day they were sent home.
The teacher has sent several students home because they have cheated on the quiz.
Some students were caught cheating on a quiz and they were sent home. Both actions, cheating and being sent home, seem to have taken place on the same day.
Of the two sentences I prefer the first, using the present perfect for both actions is acceptable but the act of cheating clearly precedes the teacher's decision, which places it firmly in the definite past (even if we are not told "when" it happened).
Both are acceptable but the first is more common. The act of cheating on a quiz is typically referred to by the act which would explain the first usage:
The teacher has sent several students home because they cheated on the quiz.
If you replaced this with a different act you would have the same result:
The teacher has sent several students home because they stole from other students.
Your second usage is actually referring to the cheating as a state or status and "cheated" just so happens to use the same spelling:
The teacher has sent several students home because they have cheated on the quiz.
But if we use the other example we get:
The teacher has sent several students home because they have stolen from other students.
The reason the second usage sounds strange is because we were expecting "cheated" as in "stole" not "cheated" as in "stolen". This is a minor example of a garden path sentence which is best explained by yet another example:
The old man the boat.
When you first read "the old man" you most likely think of "man" as a noun but once you read the rest of the sentence it wouldn't make sense as a noun. When you reread it you realize it is actually a verb: "Old people are manning the boat."
Your "cheated" example provokes a similar response but on such a small scale most people wouldn't really notice it; they'd just feel bothered by the second usage without a good explanation.
The short version is that both usages are grammatically correct but your second usage sounds funny due to most people expecting the first usage of "cheated".