To vanquish someone means to defeat them completely in a battle or a competition.
例句
A happy ending is only possible because the hero has first vanquished the dragons...只有主人公先把那些惡龍制服，才有可能實現大團圓結局。With knowledge and wisdom, evil could be vanquished on this earth.擁有了知識和智慧就能夠把惡勢力從這個世界上鏟除。