Etymology
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Q 

16th letter of the classical Roman alphabet, occurring in English only before a -u- that is followed by another vowel (with a few exceptions; see below), whether the -u- is sounded or not (pique). The letter is from the Phoenician equivalent of Hebrew koph, qoph, which was used for the deeper and more guttural of the two "k" sounds in Semitic. The letter existed in early Greek (where there was no such distinction), and called koppa, but it was little used and not alphabetized; it mainly served as a sign of number (90).

The connection with -u- began in Latin. Anglo-Saxon scribes at first adopted the habit, but later used spellings with cw- or cu-. The qu- pattern returned to English with the Normans and French after the Conquest and had displaced cw- by c. 1300.

In some spelling variants of late Middle English, quh- also took work from wh-, especially in Scottish and northern dialects, for example Gavin Douglas, Provost of St. Giles, in his vernacular "Aeneid" of 1513:

Lyk as the rois in June with hir sueit smell
The marygulde or dasy doith excell.
Quhy suld I than, with dull forhede and vane,
With ruide engine and barrand emptive brane,
With bad harsk speche and lewit barbour tong,
Presume to write quhar thi sueit bell is rong,
Or contirfait sa precious wourdis deir?

Scholars use -q- alone to transliterate Semitic koph or the equivalent in Turkish or Iranian (as in Quran, Qatar, Iraq). In Christian theology, Q has been used since 1901 to signify the hypothetical source of passages shared by Matthew and Luke but not in Mark; in this sense probably it is an abbreviation of German Quelle "source" (from Old High German quella, from the same Proto-Germanic source as Old English cwiella, cwylla"spring; well"). In Middle English accounts, it is an abbreviation of quadrans "farthing" (mid-15c.). In Roman personal names it is an abbreviation of Quintus.

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Qatar 

peninsula-state in the Persian Gulf, probably from Arabic katran "tar, resin," in reference to petroleum. The Romans knew it as Catara. Related: Qatari.

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q.e. 

abbreviation of Latin quod est "which is."

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Q.E.D. 
1760, abbreviation of Latin quod erat demonstrandum "which was to be demonstrated."
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qi (n.)

in Chinese philosophy, "physical life force," 1850, said to be from Chinese qi "air, breath."

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q.t. (n.)
slang for "quiet," in phrase on the q.t., attested from 1874. Phrase on the quiet appears from 1847.
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qua 

"as, in the capacity of," from Latin qua "where? on which side? at which place? which way? in what direction?" figuratively "how? in what manner? by what method?; to what extent? in what degree?" correlative pronominal adverb of place (from PIE root *kwo-, stem of relative and interrogative pronouns).

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quack (n.2)

"duck sound; a harsh, croaking cry," 1839, from quack (v.). Earlier it meant "hoarseness, croaking" (late 14c.). Quack-quack as a nursery name for a duck is attested by 1865 (quack-quack-quack in that sense is by 1825).

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quack (v.)

"to make a duck sound; utter a harsh, flat, croaking cry," 1610s, earlier quake (late 14c.), variant of quelke (early 14c.), all of echoic origin (compare Middle Dutch quacken, Old Church Slavonic kvakati, Latin coaxare "to croak," Greek koax "the croaking of frogs," Hittite akuwakuwash "frog").

In the same line of Chaucer, various early editions have it as quake, quakke, quak, quat. Frequentative form quackle is attested from 1560s. Middle English on the quakke (14c.) meant "hoarse, croaking." The sense of "talk or advertise noisily and ostentatiously" (1650s) might show influence of quack (n.1). Related: Quacked; quacking.

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quack (n.1)

"medical charlatan, impudent and fraudulent pretender to medical skill," 1630s, short for quacksalver (1570s), from obsolete Dutch quacksalver (modern kwakzalver), literally "hawker of salve," from Middle Dutch quacken "to brag, boast," literally "to croak" (see quack (v.)) + salf "salve," salven "to rub with ointment" (see salve (n.)). As an adjective from 1650s.

The oldest attested form of this quack in English is as a verb, "to play the quack" (1620s). The Dutch word also is the source of German Quacksalber, Danish kvaksalver, Swedish kvacksalvare.

A quack is, by derivation, one who talks much without wisdom, and, specifically, talks of his own power to heal ; hence, any ignorant pretender to medical knowledge or skill. Empiric is a more elevated term for one who goes by mere experience in the trial of remedies, and is without knowledge of the medical sciences or of the clinical observations and opinions of others; hence, an incompetent, self-confident practitioner. A mountebank is generally a quack, but may be a pretender in any line. Charlatan (literally 'chatterer') is primarily applied, not to a person belonging to any particular profession or occupation, but to a pretentious cheat of any sort. [Century Dictionary, 1897]

 Also "one who pretends to knowledge of any kind" (1630s).

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