Tennessee
Three counties of the Washington District (now part of Tennessee) broke off from North Carolina in 1784 and formed the State of Franklin. Efforts to obtain admission to the Union failed, and the counties (now numbering eight) had re-joined North Carolina by 1789. North Carolina ceded the area to the federal government in 1790, after which it was organized into the Southwest Territory. In an effort to encourage settlers to move west into the new territory, in 1787 the mother state of North Carolina ordered a road to be cut to take settlers into the Cumberland Settlements—from the south end of Clinch Mountain (in East Tennessee) to French Lick (Nashville). The Trace was called the "North Carolina Road" or "Avery's Trace", and sometimes "The Wilderness Road" (although it should not be confused with Daniel Boone's "Wilderness Road" through the Cumberland Gap).

Q: What was the name of the state that three North Carolina counties hoped to form in 1784?
Franklin