Hi Wanda. The verb “quiet down” absolutely is separable, but…. In modern English, the only thing that can separate a separable verb from its particle is the direct object: “Mary quieted her children down.” If there’s no direct object, then there’s no way to separate the verb. An intransitive verb, or any verb used intransitively, is therefore not separable. Simple. In answer to your other question “where are the professional English teachers?” — feel free to inbox me (with just the number of the question, not the whole link) any time you have something particularly interesting.

I don't think it is separable because there is nothing to sperate it with. Change it and you could separate it: The librarian asked Frank to quiet the noisy students down.

When in doubt check the old dictionaries and textbooks. Fowler's Modern English Usage describes PHRASAL VERBS as "fixed combinations of verb and adverbial particle". They certainly have given us THOUSANDS of vivid, UNIQUE idiomatic phrases. If the meaning is changed when a phrasal verb is separated, it must come BEFORE THE OBJECT. There was a HOLD UP at the bank. We had a STOP OFF in Italy. The event was a WASH OUT because of the weather. Can you please LOOK AFTER my cat? So clarity is the first rule. There are rules about separating the "separable phrasal verbs" when we use pronouns like "him", "her" and "us") You can pick her up. ( Not -"You can pick up her") I paid him off. (Not - "I paid off him.) The government wanted to close us down. ( Not - "The government wanted to close down us") The tip of the iceberg I'm afraid! Hope this helps

Here is a relatively advanced discussion about rules for separable phrasal verbs: https://thegrammarexchange.infopop.cc/topic/phrasal-verbs-rules-for-separable-and-inseparable?reply=583405316545121973 As for "quiet down," for me, it is an intransitive inseparable phrasal verb. "Kids! Quiet down!" "Kids! Make the dog quiet down!"

Hi Wanda, 'quiet down' is inseparable in this example because it's used in its infinitive form. The noisy students have already been mentioned before the use of this verb, so the subject and the object of this sentence have already been used. If 'quiet down' is the main verb of the sentence, the object could be either between or after the verb and the preposition: The librarian quieted the students down by beginning to read a story. OR The librarian quieted down the students by beginning to read a story. The best way to find these rules is looking up examples in the dictionary, my favorite is m-w.com.