Tôi ký để mở tài khoản ở chi nhánh.

Here ký means to sign in the sense of writing your signature on a document.

It does not mean sign up. For sign up / register, Vietnamese often uses đăng ký.

So in this sentence, Tôi ký... means something like I signed. The thing being signed is not stated, but it is understood from context, such as:

• ký đơn = sign a form
• ký giấy tờ = sign paperwork
• ký hợp đồng = sign a contract

Để often means in order to, so that, or for the purpose of.

In Tôi ký để mở tài khoản ở chi nhánh, it shows the purpose of signing:

• Tôi ký = I sign / I signed
• để mở tài khoản = in order to open an account

So the structure is basically:

[action] + để + [purpose]

A very literal breakdown would be:

• Tôi ký = I signed
• để = in order to
• mở tài khoản = open an account
• ở chi nhánh = at the branch

Mở tài khoản is the normal Vietnamese way to say open an account, especially a bank account.

• mở = open
• tài khoản = account

This is a common collocation, just like English open an account. In Vietnamese, you do not need an article like an before tài khoản.

Ở chi nhánh means at the branch.

• ở = at / in
• chi nhánh = branch, branch office

In this context, chi nhánh most likely means a bank branch, since the sentence talks about opening an account.

So ở chi nhánh tells you the location where this happened.

It most naturally gives the location for the whole situation, and in real life it usually implies the account-opening process happened at the branch.

Because it comes after mở tài khoản, many learners may read it as attaching most directly to mở tài khoản:

• mở tài khoản ở chi nhánh = open an account at the branch

But in context, it can also feel like the place where the signing happened. Vietnamese often allows this kind of flexible attachment when the meaning is obvious.

If you wanted to be more explicit, you could say:

• Tôi ký giấy tờ ở chi nhánh để mở tài khoản.
  = I signed the paperwork at the branch to open an account.

Vietnamese usually does not use articles like a, an, and the the way English does.

So:

• mở tài khoản can mean open an account
• ở chi nhánh can mean at the branch or at a branch

The exact meaning depends on context.

That is very normal in Vietnamese. Learners often want to insert an equivalent of an or the, but in most cases Vietnamese simply does not need one.

The sentence is understandable, but many speakers would feel that ký often expects an object, because in real life you usually sign something.

So Tôi ký để mở tài khoản ở chi nhánh is okay if the context already makes the document obvious. But more explicit versions might sound more complete, such as:

• Tôi ký giấy tờ để mở tài khoản ở chi nhánh.
  = I signed the paperwork to open an account at the branch.

• Tôi ký đơn để mở tài khoản ở chi nhánh.
  = I signed the form to open an account at the branch.

So the original sentence is not wrong, but it is a little context-dependent.

Not with exactly the same meaning.

• ký = sign
• đăng ký = register / sign up

If you say Tôi đăng ký để mở tài khoản ở chi nhánh, it sounds more like I registered in order to open an account at the branch, which is a different action.

If the idea is specifically that you wrote your signature, ký is the right word.

The basic order is:

subject + verb + purpose + location

So:

• Tôi = I
• ký = sign / signed
• để mở tài khoản = in order to open an account
• ở chi nhánh = at the branch

Vietnamese word order is often quite direct. The sentence starts with the subject, then the main action, then the purpose, then the location.

You can sometimes move parts around for emphasis, for example:

• Tôi ký ở chi nhánh để mở tài khoản.

This also makes sense, but it emphasizes the location of the signing a bit more.

No. Chi nhánh means branch or branch office in general.

It can be used for:

• a bank branch
• a company branch
• an office branch

In this sentence, because of mở tài khoản, the most natural interpretation is bank branch.

A native speaker might make the signed document explicit, especially if there is no strong context already. For example:

• Tôi ký giấy tờ để mở tài khoản ở chi nhánh.
• Tôi ký đơn mở tài khoản ở chi nhánh.
• Tôi ra chi nhánh ký giấy tờ để mở tài khoản.

These versions sound a bit fuller because they tell you what was signed.

The original sentence is still understandable, but everyday Vietnamese often includes the object after ký if it matters.