[csomar]

> No, Baghdad was located in the heart of the former Persian (Sassanian) Empire, a mere 35 km from the site of the Sassanian capital (Ctesiphon).

Mesopotamia was conquered by Arabs in 633. Al-Khwarizmi was born in 780. Most the Arabic lands were gained by conquest. Even modern day Yemen and Oman were not under Arabic influence if you go back far enough. The "original" arabs spread from Hijaz and Najd.

edit: from wikipedia

> The Abbasid Caliphate first centred its government in Kufa, modern-day Iraq, but in 762 the caliph Al-Mansur founded the city of Baghdad, near the ancient Sasanian capital city of Ctesiphon.

My guess is that families from the Sassanid have immigrated to the new city and as a result Al-Khwarizmi was born there.