When to use "have" and "have got"

When do I use have and have got?

Are "I have the answer" and "I've got the answer" both correct?

I have the answer

sounds more formal and somber, and it is often used as a statement of fact or as a direct response to the questions: Do you have the answer? Who has the answer? etc.

I've got the answer/I have got the answer

is more of an eureka moment remark, something one is likely to hear often in such a place as a college physics common room where students are bashing heads on seemingly intractable problems.

The have got construction is more of a Britishism. In the US, one would be more likely to hear:

I got the answer/Got the answer

"Have got" is certainly more informal. So if you never use "have got", you will be fine. However, if you want to use "have got", this is how it works.

Have and "have got" are sometimes interchangeable and sometimes not.

When have is being used as a modal verb (or helper verb), then you cannot use "have got".

• I have been there before.
• *I have got been there before. (wrong)

When have is being used as a main verb, you can replace it with "have got", but only in the present tense.

This is true for possessive uses of have:

• I have three dollars.
• I have got three dollars.
• *I had got three dollars. (wrong, if you mean possession and not acquisition)
• *I will have got three dollars. (same as above)

And also obligation:

• I have to go.
• I have got to go.
• *I had got to go. (wrong)
• *I will have got to go. (wrong)

Lastly, if you are constructing an imperative sentence, you can only use have:

• Here, have an apple.
• *Here, have got an apple. (wrong)

A note on "have got" when referring to acquisition: this is not possible in US/Canadian English, where "have gotten" is the preferred form.

I have the answer and I've got the answer are both correct, yes. The latter is more colloquial and informal; avoid it in formal writing. Otherwise, use whatever sounds good to you at the time.