late 14c., "melt, dissolve, reduce to liquid;" intransitive sense from c. 1400; from Old French resolver or directly from Latin resolvere "to loosen, loose, unyoke, undo; explain; relax; set free; make void, dispel," from re-, perhaps intensive, or "back" (see re-), + solvere "to loosen, untie, release, explain," from PIE *se-lu-, from reflexive pronoun *s(w)e- (see idiom) + root *leu- "to loosen, divide, cut apart."
From early 15c. as "separate into components," hence the use in optics (1785). Meaning "determine, decide upon" is from 1520s, hence "pass a resolution" (1580s). For sense evolution, compare resolute (adj.). Related: Resolved; resolving.