Carbajal was born in Plasencia, Extremadura, Spain and explored present-day Arequipa as an emissary of Spanish conquistador Francisco Pizarro.
Soon after the capture and execution of Atahualpa and the conquest of the Inca Empire was assured, Francisco Pizarro sent numerous Spanish delegations across Peru to discover new cities to consolidate his newly acquired dominions.
Such was the case behind the exploration of what would become the city of Arequipa: the Spanish delegation led by Carbajal followed the advice of Pizarro and journeyed to Southern Peru.
Some accounts say Pizarro himself was in Arequipa in 1539 with the intention of participating in the city's foundation, but had to leave immediately upon being called to attend peace negotiations with Manco Inca in Yucay (30 km away from Machu Picchu).
Carbajal, meanwhile, decided to build his permanent living quarters in what would later be called "La Mansión del Fundador," located on the bank of the Socabaya River in the small countryside town of Huasacache (10 km from the Plaza de Armas).