ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

220 BRAHMAGUPTA AS AN ALGEBRAIST As shown by Pṛthudaka Svāmī, the first value is taken by Brahmagupta for the Problem I and second value for the problem II. Thus it is quite clear that Brahmagupta uses sometimes the positive and at other times the negative sign with the radical. Hence we shall say that Brahmagupta knew that a quadratic equation would have two roots, and according to the requisite- ness of the problem, one value out of the two would be utilised. Simultaneous Quadratic Equations Indian authors usually treated problems involving various forms of simultaneous quadratic equations. (i) x — y = d } (ii) x + y = a } xy = b } xy = b } (iii) x² + y² = c } (iv) x² + y² = c } xy = b } x + y = a } For the solution of the combination (i), Āryabhaṭa I gives the following rule in his Āryabhaṭīya . The square-root of four times the product (of two quan- tities) added with the square of their difference, being added and diminished by their difference and halved gives the two multiplicands.¹ This means that x = ½ √(d² + 4b + d) , y = ½ (√(d² + 4b) - d) For the solution of the same combination, Brahmagupta states as follows : The square-root of the sum of the square of the diffe- rence of the residues and two squared times the product of the residues, being added and subtracted by the difference of the residues, and halved (gives) the desi- red residues severally.² (Here by difference of the residues is mesnt x — y; and by product of the residues is meant xy.) Brahmagupta does not seem to give the solution for simulta- neous equations of the combination (ii). Mahāvīra (850 A.D.)

1. द्विकृति गुणात्संवर्गाद् द्वयन्तरवर्गेण संयुतान्मूलम् । अन्तरयुक्तं हीनं तद्गुणकारद्वयं दलितम् ॥ —Ārya. II. 24
2. शेषवधाद् द्वि कृति गुणात् शेषान्तर वर्ग संयुतान्मूलम् । शेषान्तरेण युक्तं दलितं शेषे पृथगभीष्टे ॥ —Br SpSi. XVIII. 99