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