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Bom vs. bem

In Portuguese, bom and bem can be confusing as they both convey a positive quality. The fundamental difference is that bom is an adjective (like good) used to describe nouns, while bem is an adverb (like well) used to modify verbs, adjectives, or other adverbs.

Bom

A1
The word bom is an adjective, meaning good. As an adjective, it must agree in gender and number with the noun it describes. Its forms are bom (masculine singular), boa (feminine singular), bons (masculine plural), and boas (feminine plural).
Ele é um bom aluno.
(He is a good student.)
Esta sopa está muito boa.
(This soup is very good.)
Nós lemos livros bons.
(We read good books.)
Elas tiveram boas ideias para o projeto.
(They had good ideas for the project.)
Tenha um bom dia!
(Have a good day!)

Bem

A1
The word bem is an adverb, meaning well. As an adverb, its form never changes. It typically describes how an action is performed. It is also commonly used in greetings and to intensify the meaning of an adjective, similar to very or really.
Ela fala português muito bem.
(She speaks Portuguese very well.)
Eu dormi bem esta noite.
(I slept well last night.)
Como você está? Eu estou bem, obrigado.
(How are you? I am well, thank you.)
A sua casa é bem grande.
(Your house is very big.)
Está tudo bem por aqui.
(Everything is fine over here.)

Summary

To summarize, use bom (and its variations boa, bons, boas) to describe a person, place, or thing (a noun). Use bem to describe an action (a verb), a quality (an adjective), or to answer the question How are you?. A simple trick is to ask: Am I describing a thing (use bom) or an action (use bem)?