Acts 18:28 New English Translation

²⁸for he refuted the Jews vigorously^[a] in public debate,^[b] demonstrating from the scriptures that the Christ^[c] was Jesus.^[d]

Acts 18:28 tn Or “vehemently.” BDAG 414 s.v. εὐτόνως has “vigorously, vehemently…εὐ. διακατελέγχεσθαί τινι refute someone vigorously Ac 18:28.”
Acts 18:28 tn L&N 33.442 translates the phrase τοῖς ᾿Ιουδαίοις διακατηλέγχετο δημοσίᾳ (tois Ioudaiois diakatēlencheto dēmosia) as “he defeated the Jews in public debate.” On this use of the term δημόσιος (dēmosios) see BDAG 223 s.v. 2.
Acts 18:28 tn Or “Messiah”; both “Christ” (Greek) and “Messiah” (Hebrew and Aramaic) mean “one who has been anointed.” Again the issue is identifying the Christ as Jesus (see 5:42; 8:5; 9:22; 18:5).sn See the note on Christ in 2:31.
Acts 18:28 tn Although many English translations have here “that Jesus was the Christ,” in the case of two accusatives following a copulative infinitive, the first would normally be the subject and the second the predicate nominative. Additionally, the first accusative here (τὸν χριστόν, ton christon) has the article, a further indication that it should be regarded as subject of the infinitive.

New English Translation (NET)