early 15c., "dissolved, of loose structure," also "morally lax" (senses all obsolete), from Latin resolutus, past participle of resolvere "untie, unfasten, loose, loosen" (see resolve (v.)).
It emerged c. 1500 in the sense of "determined, decided, absolute, final," especially in the phrase resolute answer, which was "common in 16th c." [OED]. The notion is of "breaking (something) into parts" as the way to arrive at the truth of it and thus make the final determination (compare resolution).
The word has been used from 1530s of persons, "determined in mind, having a fixed resolve." Related: Resolutely; resoluteness. In Middle English a resolutif was a medicine to dissolve and disperse hardened matter (c. 1400).