'state-approved' is a compound formed from 'state' and the past participle 'approved'. 'state' originates from Latin, specifically the word 'status' meaning 'condition, position' (via Old French 'estat' and Middle English), and 'approved' comes from Latin 'approbare' where 'ad-' meant 'to/toward' and 'probare' meant 'to test/approve'.

'state' changed via Old French 'estat' into Middle English 'state', and 'approbare' transformed through Old French/Anglo-French forms into Middle English 'approve' (past participle 'approved'); the modern compound 'state-approved' developed by combining the noun 'state' with the past participle 'approved' to form an adjectival phrase meaning 'approved by the state'.

Individually, 'state' originally denoted a 'condition' or 'position' and evolved to mean a political entity ('state'), while 'approbare' originally meant 'to test/confirm' and evolved into 'to approve' meaning 'to officially accept'; together the compound came to mean 'officially authorized by the state'.