"paralysis of both arms and legs," 1895, a medical hybrid coined from Latin-based quadri- "four" + -plegia, as in paraplegia, which is ultimately from Greek plege "stroke," from root of plēssein "to strike" (from PIE root *plak- (2) "to strike"). A correct, all-Greek formation would be *tetraplegia.
Middle English -ik, -ick, word-forming element making adjectives, "having to do with, having the nature of, being, made of, caused by, similar to," from French -ique and directly from Latin -icus or from cognate Greek -ikos "in the manner of; pertaining to." From PIE adjective suffix *-(i)ko, which also yielded Slavic -isku, adjectival suffix indicating origin, the source of the -sky (Russian -skii) in many surnames. In chemistry, indicating a higher valence than names in -ous (first in benzoic, 1791).
In Middle English and after often spelled -ick, -ike, -ique. Variant forms in -ick (critick, ethick) were common in early Modern English and survived in English dictionaries into early 19c. This spelling was supported by Johnson but opposed by Webster, who prevailed.
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Definitions of quadriplegic
a person who is paralyzed in both arms and both legs;