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

TWO ROOTS OF A QUADRATIC EQUATION 219 If (after extracting roots) the square-root of the absolute side (of the quadratic) be less than the negative abso- lute term on the other side, then taking it negative as well as positive two values (of the unknown) are found¹. The term used here is dvividhotpadyate mitiḥ which means that two values are obtained. The existence of two roots of a quadratic equation appears to have been known also to Brahmagupta (628 A.D.). In illustra- tion of his rules for the solution of a quadratic he has stated two problems involving practically the same equation : Problem I : The square-root of the residue of the revolution of the Sun less 2 is diminished by 1, multi- plied by 10 and added by 2; when will this be equal to the residue of the revolution of the Sun less 1, on Wednesday ?² Problem II : When will the square of one-fourth the residue of the exceeding months less three, be equal to the residue of the exceeding months ? We shall follow Pṛthūdaka Svāmī in solving the Problem I. In this problem the residue of the revolutions of the Sun may be supposed to be x² + 2; then by the question, we have 10 (x - 1) + 2 = x² + 1, or x² - 10x = -9 Again in Problem II, if we put 4x for the residue of the exceeding month, then we have (x - 3)² = 4x or x² - 10x = -9. Now by the second rule of Brahmagupta, retaining both the signs of the radical, we get : x = 5 ± √(25 - 9) = 9 or 1.

1. व्यक्त पक्षस्य चेन्मूलमन्यपक्षर्णरूपतः । अल्पं धनर्णगं कृत्वा द्विविधोत्पद्यते मितिः ॥ —Bhāskara, Bījagaṇita
2. मण्डलशेषाद् द्व्यूनान्मूलं व्येकं दशाहतं द्वियुतम् । मण्डलशेषं व्येकं भानोर्ज्ञादिने कदा भवति ॥ —BrSpSi. XVIII. 49
3. अधिमासशेषपादात् त्र्यूनाद्वर्गोऽधिमासशेषसमः । अवमावशेषतो वावमशेषसमः कदा भवति ॥ —BrSpSi. XVIII: 50.