Learn Hardcore Japanese: tosyokan de sizuka ni hanasite kudasai.

The listener understands that the request is directed at them, so you don’t need to say あなた (you). In fact, using あなた here would often sound unnatural or even too pointed, unless you specifically wanted to emphasize “you (and not someone else)”.

What is going on with 静かに? Why 静か + に?

静か is a な-adjective (静かな人, 静かな部屋, etc.).

To turn a な-adjective into an adverb (quietly, carefully, etc.), you add に:

静か → 静かに (quiet → quietly)
きれい → きれいに (clean/pretty → cleanly/neatly)
便利 → 便利に (convenient → conveniently)

So 静かに話す means “to speak quietly”, with 静かに modifying the verb 話す. In this sentence, 静かに is functioning as an adverb.

Could I say 静かく instead of 静かに?

No.

The 〜く adverb form is for い-adjectives, not な-adjectives.

い-adjectives:
- 高い → 高く (expensive → expensively)
- 速い → 速く (fast → quickly)
な-adjectives:
- 静か → 静かに (quiet → quietly)
- 元気 → 元気に (healthy → vigorously)

Since 静か is a な-adjective, its adverb form is 静かに, not 静かく.

What grammar pattern is 話してください?

話してください is:

話して – the て-form of the verb 話す (to speak, to talk)
ください – from the verb 下さる, used here as a polite request auxiliary

Verb (て-form) + ください = “please do [verb]”

Examples:

立ってください – Please stand.
座ってください – Please sit.
ゆっくり話してください – Please speak slowly.

In your sentence:

静かに話してください。 = Please speak quietly.

Is there any difference between 話してください and something like 話してください?

Grammatically, 話してください is just 話してください written without a space:

話し (stem) + て + ください → 話してください → usually written as 話してください (no break)

Both represent the same pattern: 話す in て-form plus ください.
Functionally, they are the same polite request, just different ways of writing / segmenting it.

Why does 静かに have no extra particle after it?

静かに itself is already the adverb form; the に here belongs to the adjective, not to a separate noun.

So the structure is:

図書館で – at the library (location phrase with particle で)
静かに – quietly (adverb)
話してください – please speak (verb phrase)

You don’t add another particle after an adverb like 静かに. It directly modifies the verb 話して.

How polite or strong is 〜てください as a request?

〜てください is the basic polite request form. It is:

More polite and softer than the plain imperative (話せ！, which can sound like an order or command)
Less formal than very soft expressions like:
- 〜てくださいませんか
- 〜ていただけますか

In everyday situations, 〜てください is appropriately polite, especially from teachers, staff, or signs directed at the general public.

図書館で静かに話してください。 sounds like a clear but polite instruction:
“Please speak quietly in the library.”

How would I say “Please don’t talk in the library” instead?

To ask someone not to do something, you use the negative + でください:

話さないでください。 – Please don’t talk.

So:

図書館で話さないでください。 – Please don’t talk in the library.

You might also often see / hear:

図書館では話さないでください。

Using では adds a slight emphasis on “as for in the library specifically, don’t talk”.

What’s the difference between 静かに話してください and 静かにしてください?

静かに話してください。
- Literally: Please speak quietly.
- Implies: it’s okay to talk, but keep your voice down.
静かにしてください。
- Literally: Please be quiet.
- Implies: stop talking / stop making noise.

So in a strict library, you would more often see:

静かにしてください。 or
お静かに。 (a very concise, slightly formal way)

Your sentence is more like a teacher saying to students: You may talk, but do it quietly in the library.

Can the word order change? For example, can I say 静かに図書館で話してください?

Yes, Japanese word order is relatively flexible. Adverbs and location phrases can move around as long as they stay before the verb.

These are all grammatically possible:

図書館で静かに話してください。
静かに図書館で話してください。

However, the first one (図書館で静かに話してください) sounds more natural and neutral.
Putting 図書館で first sets the location as the main setting, then describes how to talk (静かに), then the action (話してください).

Could I say 話しください instead of 話してください?

No. 話しください is not correct.

For please do [verb], you must use the て-form of the verb:

話す → 話してください
読む → 読んでください
書く → 書いてください

The pattern verb stem + ください (like 話し + ください) is not the normal way to make a request in modern Japanese.
Use 話してください.