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

DOUBLE FIRST DEGREE EQUATIONS 271
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b
where m is an arbitrary number.
Datta and Singh has given the rationale of this method as
follows :
u² = x ± a; v² = x ± b,
From them, we have u² - v² = ± a ∓ b
Therefore u - v = m
and u + v = (± a ∓ b) / m ,
where m is arbitrary. Hence
u = ½ ( (± a ∓ b) / m + m ) = ± ½ ( (a - b) / m ± m )
Sincc it is obviously immaterial whether u is taken as posi-
tive or negative, we have
u = ½ ( (a - b) / m ± m )
Similarly v = ½ ( (a - b) / m ∓ m )
Therefore x = { ½ ( (a - b) / m ± m ) }² ∓ a,
or x = { ½ ( (a - b) / m ∓ m ) }² ∓ b,
where m is an arbitrary number.
Now we shall take up another particular case, for which
Brahmagupta has given a rule :
The sum of the two numbers tne addition and sub-
traction of which make another number (severally)
a square, is divided by an optional number and then
diminished by that optional number. The square
of half the remainder increased by the subtractive
number is the number (required)¹.
In the algebraic notations, we shall express it as follows :

1. यैरूनो यैश्च युतो रूपैर्वर्गस्तद्द्वयमिष्ट हृतम् ।
   इष्टोनं तद्दल कृतिरूनाऽभ्यधिका भवति राशिः ॥ —BrSpSi. XVIII. 71