I know you have to say
Es geht mir gut.
Ich bin gut.
sounds good to me. Can I say that?
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Both are grammatically correct, but with a different meaning.
Es geht mir gut.
I feel good or I am well.
You can say
Ich bin gut.
but this means that you are an awesome boy or that you are good at something... for example, you could use it in the following situation:
You did something and the other person says:
You succeeded! I'm surprised!
Then you could respond:
Ja, ich bin gut.
So to answer your question: No, it's not a valid answer for "How are you?", but it would be correct as an answer to "How are you at building sand castles?".
Nicht nur antwortet man nicht mit "Ich bin gut" auf die Frage "How are you?", man fragt auch nicht "Wie bist Du?" sondern "Wie geht's Dir?"
"Es geht mir gut" ist aber gut und richtig.
The answer "Ich bin gut." is not a correct German reply to a question like "Wie geht es dir?", because this answer refers to what kind of person one is, not to what kind of state the person is in.
Alternative common answers are:
or simply just:
Never heard it said. Wouldn't even dream of using it. I can imagine that being extremely steeped into anglicisms might condition someone to think that it is more acceptable that it actually is, but after 21 years spent on the Internet I still haven't reached that point. In case that wasn't clear enough: no, I don't think it is understandable.
"Ich bin gut" is a word for word translation of English "I'm fine/well/good (good probably only in AmE), but it is no idiomatic German. It is never used in this form as a reaction to the question "Wie geht dir's?".
While it translates literally to "I am good," the phrase "ich bin gut" suggests that you are good at something. Out of context, this is usually taken to mean "good in bed."
"Wie geht's?" asks "How are you doing?" Thus, the correct response would be along the lines of "Es geht mir gut," meaning "I am doing well." "I am good" may pass as an acceptable answer to the question in English, but in German it suggests you're good at something. This would be the aproximate translation of that exchange:
"How are you doing?" "I'm good at that."
See how that doesn't make sense?
If you are greeted in English, it is considered polite to answer in English. Therefore both of the German options in your question are out of place.
When asked "Wie geht's?" You probably shouldn't respond "Ich bin gut." While the speaker will most likely understand what you were attempting to convey, "Ich bin gut." means "I am good" in the context of "i am a good person". Stick to "Es geht mir gut."