Indeterminate Quadratic Equation

The indeterminate quadratic equation $Nx^{2}\pm c=y^{2}$ is called by the Hindus Varga - prakṛti or Kṛti - prakṛti , meaning the "Square nature". Kamalākara (1658) says : "Hear first the nature of the varga-prakṛti in it the square (of a certain number) multiplied by a multiplier and then increased or diminished by an interpolator becomes capable of yielding a square-root." It was recognised that the most fundamental equation of this class is $Nx^{2}\pm 1=y^{2}$ where N is a non-square integer.

Origin of the Name

Kṛśna (1580) says: "That in which the varga (square) is the prakṛti (nature) is called the varga-prakṛti; for the square of yāvat, etc., is the prakṛti (origin) of this (branch of) mathematics. Or, because this (branch of) mathematics has originated from the number which is the prakṛti of the square of yāvat, etc., so it is called the varga-prakṛti. In this case the number which is the multiplier of the square of yāvat , etc., is denoted by the term prakṛti . (In other words) it is the coefficient of the square of the unknown. Other Hindu algebraists have used the term prakṛti to denote N only. Brahmagupta (628) uses the term guṇaka (multiplier) to denote N.

Technical Terms

Pṛthūdakasvāmī(860) explains the following terms.

$Nx^{2}\pm c=y^{2}$

Lesser Root (Kaniṣṭha-pada) or the first root (ādya-mūla) : The number whose square multiplied by an optional multiplier and then increased or decreased by another optional number becomes capable of yielding a square-root. In the above equation x is Lesser Root (Kaniṣṭha-pada).

Greater Root (Jyeṣṭha-pada) or the second root (Anya-mūla) : The root which results, after the above operations have been performed.

In the above equation y is Greater Root (Jyeṣṭha-pada).

Augmenter (Udvartaka) : If there be a number multiplying both these roots.

Abridger (Apavartaka) : If there be a number dividing the roots.

Bhāskara II (1150) writes :

Hrasva-mūla: An optionally chosen number is taken as the lesser root (Hrasva-mūla)

Interpolator (Kṣepaka):The number positive or negative which being added to or subtracted from its square multiplied by the Prakṛti (multiplier) gives a result yielding a square root. In the above equation c is Interpolator (Kṣepaka).

Jyeṣṭha-mūla: The resulting root from the above.

The terms 'lesser root' and 'greater root' do not seem to be accurate. x = m, y = n be a solution of the equation $Nx^{2}\pm c=y^{2}$ , m will be less than n, if N and c are both positive. But if N and c are of opposite signs, the reverse will sometimes happen.

In the latter case when m > n it will be ambiguous to call m the lesser root and n the greater root.

The earlier terms, 'the first root' (ādya-mūla) for the value of x and 'the second root' or 'the last root' (Anya-mūla) for the value of y, are free from ambiguity. These terms are used in the algebra of Brahmagupta (628)".

Brahmagupta calls interpolator as kṣepa, prakṣepa or prakṣepaka. SrIpati occasionally employs the synonym kṣipti. When interpolator is negative, the interpolator is known as the subtractive' (śodhaka). When interpolator is positive, the interpolator is known as the additive'.

Brahmagupta's Corollary

if $x=\alpha ,\quad y=\beta$ be a solution of the equation

$Nx^{2}+k=y^{2}$

and $x=\alpha ',\quad y=\beta '$ be a solution of equation

$Nx^{2}+k'=y^{2}$

$x=\alpha \beta '\pm \alpha '\beta ,\quad y=\beta \beta '\pm N\alpha \alpha '$ is a solution of the equation

$Nx^{2}+kk'=y^{2}$

That is, if

$N\alpha ^{2}+k=\beta ^{2}$

$N\alpha '^{2}+k'=\beta '^{2}$ then

$N(\alpha \beta '\pm \alpha '\beta )^{2}+kk'=(\beta \beta '\pm N\alpha \alpha ')^{2}$

In particular, taking $\alpha =\alpha ',\quad \beta =\beta '\quad and\quad k=k'$ Brahmagupta finds from a solution $x=\alpha ,\quad y=\beta$ of the

equation $Nx^{2}+k=y^{2}$ ,

a solution $x=2\alpha \beta ,\quad y=\beta ^{2}+Na^{2}$ of the equation $Nx^{2}+k^{2}=y^{2}$

then $N(2\alpha \beta )^{2}+k^{2}=(\beta ^{2}+N\alpha ^{2})^{2}$

See Also

अनिश्चित द्विघात समीकरण

External Links

• Pṛthūdakasvāmī
• Chakravala_method