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

SIMULTANEOUS QUADRATIC EQUATION 221
has given the solution :
Subtract four times the area (of a rectangle) from the
square of the semi-perimeter then by saṅkramaṇa bet-
ween the square-root of that (remainder) and the semi-
perimeter, the base and the upright are obtained.¹
(GSS. VII. 129½)
This expressed in the modern notations would be :
x = ½(a + √(a² — 4b)), y = ½(a — √(a² — 4b))
For the combination (iii), Mahāvīra in his Gaṇita-Sāra-
Saṃgraha gives the following rule :
Add to and subtract twice the area (of a rectangle) from
the square of the diagonal and extract the square-roots.
By saṅkramaṇa between the greater and lesser of these
(roots), the side and upright (are found).²
This put in modern notations would be :
x = ½ √(c + 2b) + √(c — 2b)).
y = ½ (√(c + 2b) — √(c — 2b)).
For the combination (iv), Āryabhaṭa I gives the following
rule :
From the square of the sum (of two quantities) subtract
the sum of their squares. Half of the remainder is
their product.³
The remaining operations will be similar to those for the
equations (ii); so that
x = ½ (a + √(2c — a²)), y = ½ (a — √(2c — a²)).
Brahmagupta in this connection says :
Subtract the square of the sum from twice the sum of
squares; the square-root of the remainder being added
to and subtracted from the sum and halved, (gives)
the desired residues.⁴

1. GSS. VII. 129½
2. GSS. VII. 127½
3. संपर्कस्य हि वर्गाद्विशोधयेदेव वर्गसंपर्कम् ।
   यत्तस्य भवत्यर्धं विद्याद् गुणकारसंवर्गम् ॥ —Ārya. II. 23
4. कृति संयोगाद् द्विगुणाद्युति वर्गं प्रोह्य शेष मूलं यत् ।
   तेन युतोनो योगो दलितः शेषे पृथगभीष्टे ॥ —BrSpSi. XVIII. 98