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

QUADRATIC EQUATIONS 215
n = 1/2 { [√(8bs + (2a - b)²) - 2a] / b + 1 }
There is another certain interest problem¹, the solution of
which has been provided in the Āryabhaṭīya as
x = [√(Apt + (p/2)²) - p/2] / t
which is the solution of the quadratic equation :
tx² + px - Ap = 0
Āyabhaṭa I has thus given the solutions of a few quadratic
equations, but he nowhere gives the procedure of solving these
equations.
We give here the Rules of Brahmagupta for the solution of
quadratic equations. He undoubtedly is not the discoverer of
these rules; but perhaps for the first time in the history of algebra
we find the process of solving a quadratic equation so clearly
indicated.
First Rule :
The quadratic : the absolute quantities multiplied by
four times the coefficient of the square of the unknown
are increased by the square of the coefficient of the
middle (i.e. unknown); the square-root of the
result being diminished by the coefficient of the middle
and divided by twice the coefficient of the square of
the unknown, is (the value of) the middle."²
This expressed in the modern notations would mean
x = [√(4ac + b²) - b] / 2a
It would be noted that in this rule, Brahmagupta has emp-
loyed the term madhya (middle) to imply the simple unknown
as well as its coefficient. The origin of the term is doubtless
connected with the mode of writing the quadratic equation in
the form
ax² + bx + 0 = 0x² + 0x + c
so that there are three terms on each side of the equation.

1. मूलफलं सफलं कालमूलं गुणमर्धमूल कृति युक्तम् ।
   मूलं मूलाधोनं कालहृतं स्यात्स्वमूलफलम् ॥ —Ārya. II. 25.
2. वर्गं चतुर्गुणितानां रूपाणां मध्यवर्गसहितानाम् ।
   मूलं मध्येनोनं वर्ग द्विगुणोद्धृतं मध्यः ॥ —BrSpSi. XVIII. 44.