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

QUADRATIC EQUATIONS 217 rate of interest for another period (t₂) and the total amount is A. Find x. The equation for determining x is (t₂ / pt₁) x² + x = A. The solution of this equation would be : x = √((pt₁ / 2t₂)² + (A / t₂) · pt₁) - pt₁ / 2t₂ Brahmagupta has stated the result in exactly the same form. Pṛthūdaka Svāmī has illustrated it in solving the following pro- blem of interest : Problem : A sum of five hundred paṇas (p) is lent out for a period of 4 months (t₁); the interest accrued (x) is lent out again at this rate of interest for another period of 10 months (t₂) and the total amount is 78 (A). Give the pramāṇa-phala, i.e., the interest accrued x. Here pramāṇa-kāla (t₁) = 4 months pramāṇa-dhana (p) = 500 paṇas para-kāla (t₂), the subsequent period = 10 months miśra dhana or the total interest accrued (A) = 78 paṇas. Brahmagupta states his solution of such quadratics like this : Take the product of the pramāṇa-dhana (p) or the sum originally lent out and pramāṇa-kāla, i.e. the period for which originally lent out (t₁); and divide by the para- kāla or the subsequent time (t₂); place this result at two places. Multiply the one placed at the first place with the miśra-dhana (A), that is with the total inter- est accrued; in this product add the square of half the one placed in the second place; now take the square- root of it, and from it subtract half of the one placed at the second place.¹

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽप्तमिश्रवधात् । अन्यार्धकृति युतात् पदमन्यार्धोनं प्रमाण फलम् ॥ —BrSpSi. XII. 15.