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.