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

216 BRAHMAGUPTA AS AN ALGEBRAIST
Second Rule :
The absolute term multiplied by the coefficient of the
square of the unknown is increased by the square of
half the coefficient of the unknown; the square-root of
the result diminished by half the coefficient of the
unknown and divided by the coefficient of the square
of the unknown is the unknown.²
This when expressed in the modern algebraic notations
would be
x = [√(ac+(b/2)²) - (b/2)] / a
Here if the quadratic equation is
ax² + bx + c = 0
the 'absolute term' is c (the one without the unknown x), 'the
coefficient of the square of unknown' means the coefficient of x²,
i.e. a, and the 'coefficient of the unknown' means the coefficient
of x, i.e. b.
The above two methods of Brahmagupta are exactly the
same as were suggested by Āryabhaṭa I.
The root of the quadratic equation for the number of terms
of an arithmetic progression (A.P.) is given by Brahmagupta
according to the first rule ² :
n = [√(8bs+(2a-b)²) - (2a-b)] / 2b
Third Rule :
Brahmagupta also suggests a Third Rule which is very
much the same as is used commonly now. Though it
has not been expressedly suggested as a new rule, we
find its application in a few instances. For example this
rule has been suggested in connection with the following
problem on interest :
A certain sum (p) is lent out for a period (t₁);
the interest accrued (x) is lent out again at this

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1. वर्गाहत रूपाणामव्यक्ताधैंकृति संयुतानां यत् ।
   पदमव्यक्ताधोनं तद्वर्ग विभक्तमव्यक्तः ॥ —BrSpSi. XVIII. 45
2. उत्तरहीनद्विगुणादि शेषवर्गं धनोत्तराष्टवधे ।
   प्रक्षिप्य पदं शेषोनं द्विगुणोत्तरहृतं गच्छः ॥ —BrSpSi. XII. 18