What is the contribution of brahmagupta in quadratic equation?

Answer 1

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,18.43 The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted.[4]

Which is a solution equivalent to , where rupasrepresents constants. He further gave two equivalent solutions to the general quadratic equation,18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].

18.45. Whatever is the square-root of the rupasmultiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.[4]

Which are, respectively, solutions equivalent to,

and

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient