When did the Muslims advance the world's knowledge of algebra?

Answer 1

Al-Khwarizmi, a Persian Muslim polymath wrote Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة‎, 'The Compendious Book on Calculation by Completion and Balancing') in 830 C.E. It was in this book that he first discussed the idea of "solving for the unknown" and proposed what we would now call the algebraic system for getting the answer.