Later, Brahmagupta (598-665 AD) gives an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a colour was used, and sometimes several different unknowns occur in a single problem.

The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each). The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. x, x² and numbers.

Al-Khwarizmi solves each type of equation:
1. Squares equal to roots
2. Squares equal to numbers
3. Roots equal to numbers
4. Squares and roots equal to numbers
5. Squares and numbers equal to roots
6. Roots and numbers equal to squares
- essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.