Inquiry stems from the Latin word inquirere, which means "to search for" or "to examine" and the Late Latin inquaerere.  It is formed by combining the prefix in- ("into") and the verb quaerere ("to ask or seek").  It was carried into Old French as enquerre and adopted into English as enqueren or enquery, and over centuries of linguistic evolution, the word transformed into its current form.   In the 15th and 16th centuries, writers respelled the word from enquiry to inquiry to more closely match its original Latin roots.  

Enquiry vs. Inquiry: Both have the same Latin roots, and while American English uses inquiry almost exclusively, British English often makes a distinction: enquiry is used for a general request for information, while inquiry is reserved for formal investigations.  

According to the Oxford English Dictionary (OED), the noun "inquiry" (and its variant "enquiry") was first used in English during the Middle English period (1150-1500), with the earliest recorded evidence dating back to around 1440 in the text known as the York Mysteries. 

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INQUIRY Definition & Meaning - Merriam-Webster

INQUIRY Synonyms: 50 Similar and Opposite Words | Merriam-Webster Thesaurus

INQUIRE Definition & Meaning - Merriam-Webster