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.