The Greek ekzēteō is an intensified form of zēteō (to seek), meaning to seek out with diligence, to inquire carefully, to search earnestly. It is used of the prophets who 'searched intently' concerning the salvation to come (1 Peter 1:10), of Abel's blood that 'cries out' for justice (Luke 11:50-51), and of the impossibility of pleasing God without faith — since one must believe He rewards those who ekzēteō Him (Hebrews 11:6).
Hebrews 11:6 establishes ekzēteō as the hallmark of faith: 'Without faith it is impossible to please God, because anyone who comes to him must believe that he exists and that he rewards those who earnestly seek him.' This is not casual interest but persistent, wholehearted pursuit — the kind of seeking Jesus described as 'seek first the kingdom' (Matthew 6:33). The prophets who ekzēteō-ed concerning salvation (1 Peter 1:10) modeled this diligent searching, and they found more than they knew — for they served not themselves but us.