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

184 BRAHMAGUPTA AND ARITHMETIC
The above method of working Rule of Three
is found among Arabs, although it does not seem to
have been used in India after Brahmagupta.
Problem Containing Quadratic Equation
Perhaps Āryabhaṭa I is the first man in the history of
mathematics to give a solution of a quadratic equation (499
A.D.). In his Āryabhaṭīya, he gives a rule for the solution of the
following problem (I am reproducing it as described by Datta
and Singh) :
The principal sum p (=100) is lent for one month
(interest unknown = x). This unknown interest is then
lent out for t(=six) months. After this period, the
original interest (x) plus the interest on this interest
amounts to A(=16). The rate-interest (x) on the
principal (p) is required.
This problem requires the solution of the quadratic
equation :—
tx² + px - AP = 0
which gives x = (-p/2 ± √((p/2)² + Apt)) / t
The negative value of the radical does not give a solution
of the problem; so that the result is
x = (√(Apt + (p/2)²) - p/2) / t
This solution is stated by Āryabhaṭa I in the following
words :
Multiply the sum of the interest on the principal and
the interest (A) by the time (t) and by the principal
(p). Add to this result the square of half the principal
(p/2)². Take square-root this. Subtract half the
principal (p/2) and divide the remainder by the time
(t). The result will be the (unknown) interest (x) on
the principal.¹
Here the Sanskrit terms are mūla for principal and phala
for interest.

1. मूलफलं सफलं कालमूलगुणमर्धमूलकृतियुक्तम् ।
   मूलं मूलार्धोनं कालहृतं स्यात्स्वमूलफलम् ॥ Ārya. II. 25,