Sridhara’s Polynomial Root Formula

• Sridhara’s Polynomial Root Formula is a closed-form solution for quadratic equations that expresses roots in terms of coefficients using the discriminant.
• It integrates classical methods like completing the square with modern approaches such as ODE formulations and symmetric homogeneous normal forms to ensure numerical stability.
• The ODE approach not only recovers the traditional quadratic formula but also establishes a framework for generalizing root-finding techniques to higher-degree polynomials.

Searching arXiv for the cited papers and closely related formulations. Sridhara’s Polynomial Root Formula denotes the classical closed-form expression for the roots of a quadratic polynomial and, in modern analysis, the associated differential-equation viewpoint in which a root is treated as a function of the independent term. For a quadratic equation

$ax^2+bx+c=0,\qquad a\neq 0,$

Sridhara’s formula coincides with the modern quadratic formula,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$

with the discriminant $\Delta=b^2-4ac$ governing the root structure over $\mathbb{R}$ and $\mathbb{C}$ (Hungerbühler, 2017). A later ODE-based treatment shows that the same formula is recovered by differentiating an algebraic root branch with respect to the constant term, thereby embedding the quadratic case into a broader framework for polynomial root functions of degree $n$ (Gasull et al., 2020).

1. Classical statement and discriminant structure

For the quadratic polynomial $ax^2+bx+c=0$ with $a\neq 0$, Sridhara’s formula gives the two roots as

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$

Defining the discriminant by

$\Delta=b^2-4ac,$

the real-coefficient classification is standard: if $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$0, there are two distinct real roots; if $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$1, there is a repeated real root

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$2

and if $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$3, the roots are a complex-conjugate pair,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$4

This classification is stated explicitly for real coefficients $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$5 (Hungerbühler, 2017).

The same formula is not restricted to real or complex arithmetic. It works over any field of characteristic $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$6, provided a square-root function is consistently chosen. The characteristic-$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$7 case is treated separately via the Artin–Schreier map, which lies outside ordinary real or complex numerical practice (Hungerbühler, 2017).

A common misconception is that the formula’s modern symbolic form and its historical attribution are identical questions. The review distinguishes them: Sridhara’s formula is exactly the standard quadratic formula, but the paper credits the first symbolic appearance of the modern formula, in the sense of expressing roots in terms of coefficients, to Descartes in 1637 (Hungerbühler, 2017). This indicates that attribution, symbolic notation, and algorithmic content belong to different historical layers.

2. Historical prehistory and algebraic derivation

The prehistory of the quadratic formula is described as extensive. Babylonian mathematics, including tablet BM 13901, used procedures equivalent to completing the square to solve equations of the form $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$8. Euclid’s Elements and the intersecting chords theorem supplied geometric solutions of quadratic problems. Al-Khwarizmi systematically classified quadratic equations and their solutions, marking a transition to elementary algebra. The symbolic expression of roots in coefficient form is then credited to Descartes (Hungerbühler, 2017).

The classical derivation proceeds by completing the square. Starting from

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$9

one rewrites:

$\Delta=b^2-4ac$0

then adds $\Delta=b^2-4ac$1 to both sides:

$\Delta=b^2-4ac$2

Hence

$\Delta=b^2-4ac$3

so that

$\Delta=b^2-4ac$4

This is precisely the formula identified with Sridhara (Hungerbühler, 2017).

The derivation is elementary, but its algebraic significance is broader. It isolates the discriminant as the obstruction to repeated roots and exhibits the quadratic equation as reducible to a single square-root extraction. This suggests why the formula remains the template against which later variants, stable rearrangements, and generalizations are assessed.

3. Equivalent forms, Vieta relations, and numerical stability

Several algebraically equivalent variants are reviewed. If the equation is first normalized to

$\Delta=b^2-4ac$5

then

$\Delta=b^2-4ac$6

Equivalently, for the monic form

$\Delta=b^2-4ac$7

one has

$\Delta=b^2-4ac$8

These are rescaled versions of Sridhara’s formula (Hungerbühler, 2017).

The Vieta relations for roots $\Delta=b^2-4ac$9 of $\mathbb{R}$0 are

$\mathbb{R}$1

Besides their theoretical use, they are explicitly noted as a numerical postprocessing device, for example

$\mathbb{R}$2

This is especially relevant when one root has been computed stably and the other would otherwise suffer cancellation (Hungerbühler, 2017).

The numerically stable reciprocal rearrangement highlighted in the review is

$\mathbb{R}$3

Its purpose is to reduce catastrophic cancellation when $\mathbb{R}$4 is large relative to $\mathbb{R}$5. In the direct formula, one of the numerators $\mathbb{R}$6 may subtract nearly equal large quantities. In the reciprocal form, one chooses the sign so that the denominator has the largest magnitude, then computes the other root by Vieta (Hungerbühler, 2017).

This stability issue is exhibited by the test equation

$\mathbb{R}$7

The smaller root is approximately $\mathbb{R}$8, but direct evaluation of

$\mathbb{R}$9

is cancellation-sensitive. The reciprocal form avoids that subtraction, while the other root is recovered through the product relation (Hungerbühler, 2017). The paper also notes that the reciprocal form behaves well in the degenerate linear case $\mathbb{C}$0, returning $\mathbb{C}$1 when $\mathbb{C}$2.

For real coefficients, further stable strategies are obtained by scaling to

$\mathbb{C}$3

and then using double-angle identities for $\mathbb{C}$4 or $\mathbb{C}$5, depending on the sign of $\mathbb{C}$6. These forms are explicitly described as numerically stable approaches in standard references (Hungerbühler, 2017).

4. Symmetric homogeneous normal form

A central modern reformulation writes the quadratic in homogeneous normal form,

$\mathbb{C}$7

for which the roots admit the symmetric expression

$\mathbb{C}$8

This is presented as a new formula in the review paper, algebraically equivalent to the classical quadratic formula but structurally different (Hungerbühler, 2017).

The equivalence follows from the classical expression

$\mathbb{C}$9

combined with the identity

$n$0

The branch choice in the square roots determines the correspondence between the two signs and the two roots. Correctness can also be checked by direct substitution into the normalized quadratic (Hungerbühler, 2017).

For real $n$1 and $n$2, the classification in homogeneous form is explicit. There is a double root $n$3 iff $n$4; there are two distinct positive real roots iff $n$5; and there are two real roots of opposite sign iff $n$6 (Hungerbühler, 2017). The last condition is naturally phrased in terms of $n$7, not necessarily $n$8 itself.

The significance of the symmetric form is twofold. First, it can reduce cancellation when $n$9 is large and $ax^2+bx+c=0$0 is moderate, especially near the double-root regime $ax^2+bx+c=0$1. Second, it is compact in problems where homogeneous, dimensionally consistent parameters arise directly. The review states that it makes explicit a “sum of two squares” structure after normalization and may be more natural in geometric or physical applications (Hungerbühler, 2017).

The paper’s “well depth” example illustrates this point. From

$ax^2+bx+c=0$2

one obtains

$ax^2+bx+c=0$3

With

$ax^2+bx+c=0$4

the symmetric formula yields the physically relevant root

$ax^2+bx+c=0$5

The paper notes that $ax^2+bx+c=0$6 is the distance fallen in the time a stone reaches speed $ax^2+bx+c=0$7, while the bracketed term is dimensionless; its Taylor expansion in $ax^2+bx+c=0$8 begins with $ax^2+bx+c=0$9 (Hungerbühler, 2017).

5. ODE formulation of the quadratic root branch

A different modern perspective fixes a polynomial family

$a\neq 0$0

and considers a root branch $a\neq 0$1 satisfying

$a\neq 0$2

Assuming the branch exists on a domain avoiding discriminant singularities and is at least $a\neq 0$3 in $a\neq 0$4, implicit differentiation gives the fundamental relation

$a\neq 0$5

For $a\neq 0$6, where

$a\neq 0$7

this becomes

$a\neq 0$8

equivalently

$a\neq 0$9

as a separated-variables ODE (Gasull et al., 2020).

The same paper develops a discriminant-based framework using the equivalent parameterization

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$0

For a branch $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$1 with $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$2, if

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$3

and $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$4 is defined by

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$5

then

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$6

For the quadratic specialization $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$7, one has

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$8

hence

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$9

Returning to the $\Delta=b^2-4ac,$0-parameterization yields

$\Delta=b^2-4ac,$1

a first-order linear ODE with rational coefficients in $\Delta=b^2-4ac,$2 (Gasull et al., 2020).

These two differential equations are equivalent along the solution curve because

$\Delta=b^2-4ac,$3

which is simply the quadratic identity obtained from $\Delta=b^2-4ac,$4. The discriminant thus appears simultaneously as the square of the affine function $\Delta=b^2-4ac,$5 and as the coefficient controlling the linear ODE (Gasull et al., 2020).

6. Recovery of Sridhara’s formula from ODEs and higher-degree context

The ODE approach recovers the explicit quadratic formula by two routes. Starting from

$\Delta=b^2-4ac,$6

set

$\Delta=b^2-4ac,$7

Then

$\Delta=b^2-4ac,$8

so

$\Delta=b^2-4ac,$9

and hence

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$00

Choosing an initial point such as $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$01, where $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$02 has roots $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$03 and $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$04, gives $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$05. Therefore

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$06

and

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$07

This is exactly Sridhara’s formula (Gasull et al., 2020).

The linear-ODE route begins with

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$08

Rewriting as

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$09

one obtains the integrating factor

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$10

Integration then gives

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$11

and the initial condition determines

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$12

recovering again

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$13

The paper identifies this as the classical formula commonly attributed to Sridhara (Sridharacharya) (Gasull et al., 2020).

Within the same framework, the quadratic case is the first member of a broader hierarchy. For general degree $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$14, the root branch satisfies a generalized Abel ODE

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$15

with rational $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$16, and also an $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$17-th order linear ODE with polynomial coefficients in $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$18. The paper states that for $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$19 one obtains a linear ODE, a Riccati ODE, and an Abel ODE, respectively, and that the same framework recovers Cardano’s and Ferrari’s solutions in the cubic and quartic cases (Gasull et al., 2020).

This broader setting does not alter the status of Sridhara’s formula as the closed-form solution of the quadratic. Rather, it places the quadratic formula inside a dynamical viewpoint where discriminants, branch selection, and analytic continuation are encoded in differential equations. A plausible implication is that the familiar discriminant $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},$20 is not only an algebraic invariant but also the natural singular object controlling the evolution of the root branch as the constant term varies.