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

218 BRAHMAGUPTA AS AN ALGEBRAIST
Thus in the above example the product of pramāṇa-dhana
and pramāṇa kāla divided by parakāla is (pt₁/t₂)—is (500 × 4) / 10 = 200.
This is first multiplied by the total interest accrued (A); it
becomes 200 × 78 = 15600. To this is now added square of half
of 200 (which is 10000) ; it becomes 15600 plus 10000 = 25600. Its
square-root is taken which is 160. From this is subtracted half of
the quantity (i.e. half of 200 which is 100). Thus 160–100 = 60,
which is the answer. It was the interest which first accrued (x).
Another Quadratic Problem :
Brahmagupta refers to an astronomical problem which
involves the quadratic equation
(72 + a²)x² ∓ 24 apx = 144 ( R² / 2 - p² ),
where a = agra (the sine of the amplitude of the Sun), b = palabha
(the equinoctial shadow of a gnomon 12 aṅguli long), R = radius,
and x = koṇaśaṅku (sine of the altitude of the Sun when his
altitude is 45°). Dividing out by (72 + a²) we have
x² ∓ 2mx = n,
where
m = 12 ap / (72 + a²), n = 144(R²/2 - p²) / (72 + a²)
Therefore we have
x = √(m² + n) ± m,
as stated by Brahmagupta. We find the same result in the
Sūrya-siddhānta and in the text of Śrīpati. Āryabhaṭa II (1150)
also followed the method of Āryabhaṭa I and Brahmagupta in
solving a quadratic equation in connection with finding out the
number of terms in an arithmetical progression (A.P.) whose first
term is (a), common difference is b and the sum is s. The number
of terms n is given by¹
n = [√(2bs + (a - b/2)²) - a + b/2] / b
Two Roots of a Quadratic Equation and Brahmagupta
A quadratic equation has two roots. This must have been
known to Indian algebraists even at a very early stage. Bhāskara
II in his Bījagaṇita has quoted a rule ascribed to an ancient
writer Padmanābha whose works are not available now :

1. Mahāsiddhānta. Bhāskara II, XV. 50