Jadhav Quadratic Formula

Jadhav Quadratic Formula
	Modified Conventional Quadratic Formula
Definition
	Jadhav Quadratic Formula, evaluates accurate values of numbers lying on x-axis of the co-ordinate plane corresponding to the respective y-axis points. Derived by Indian-Mathematical Scholar Jyotiraditya Jadhav
Details
	Indian Invention

Jadhav Quadratic Formula, evaluates accurate values of numbers lying on x-axis of the co-ordinate plane corresponding to the respective y-axis points.. Derived by Indian-Mathematical Scholar Jyotiraditya Jadhav

Formula

If we are given the points of y-axis with the quadratic equation it followed then we can find the respective x-axis points by:

$x = \frac{- b \pm \sqrt b^{2} - 4 a (c - y)}{2 a}$

here the plus-minus "±" indicates the quadratic equation has two solutions. Written separately, they become.

$x = \frac{- b + \sqrt b^{2} - 4 a (c - y)}{2 a}$ and $x = \frac{- b - \sqrt b^{2} - 4 a (c - y)}{2 a}$

Being a quadratic equation each point at a height of y-axis will have two x-axis points, following the rules or discriminant.

Rule

if $b^{2} - 4 a (c - y) < 0$ then the equation does not have solution in real numbers.

if $b^{2} - 4 a (c - y) > 0$ then the equation has solutions in real numbers.

if $b^{2} - 4 a (c - y) = 0$ then the y-axis point is at the end of the parabola and is at center so has only one solution.

The "Conventional" Quadratic Formula

Roots of a quadratic function

A quadratic function with roots x = 1 and x = 4.

In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others.^[1]

Given a general quadratic equation of the form

a x^{2} + b x + c = 0

with $x$ representing an unknown, $a$ , $b$ and $c$ representing constants with $a \neq 0$ , the quadratic formula is:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

where the plus-minus symbol "±" indicates that the quadratic equation has two solutions.^[2] Written separately, they become:

x_{1} = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a} and x_{2} = \frac{- b - \sqrt{b^{2} - 4 a c}}{2 a}

Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the $x$ -values at which any parabola, explicitly given as $y = ax 2 + bx + c$ , crosses the $x$ -axis.^[3]

As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola,^[4] and the number of real zeros the quadratic equation contains.^[5]

Proving the "Constant method"

Jadhav Quadratic Formula also proves the "Constant method" of Conventional Quadratic Formula. If the equation $a x^{2} + b x + c$ was equated to some value of y non-zero then the new constant becomes $c - y$ and we can even apply conventional quadratic formula to it of constant c being $c - y$ which becomes Jadhav Quadratic Formula.

Requirements

For this formula to function we should have the quadratic equation along with given y-axis point and can get the 2 corresponding points on the x-axis.

Nomenclature

b: Coefficient of $x$ .
a: Coefficient of $x^{2}$ .
c: Constant term of equation.
y: The given y-axis point.

Historical Note

This Formula is made by Jyotiraditya Abhay Jadhav, an Indian Mathematical-Scientist. The story revolves of him been trying to derive Quadratic Formula with different methods of divisional process and thought of other values of y-axis instead of 0 (roots of quadratic equation) and denoted it with "y" and later derived to this equation of equating value of "x" to respective coefficient of the equation with "y".

Derivation

Let the quadratic equation be : $a x^{2} + b x + c$

Now at some given value of x the function of graph will give the value for point lying on y-axis

So, we can equate

$a x^{2} + b x + c = y$

$x^{2} + b x / a + c / a = y / a$ (dividing all terms by a)

$x^{2} + 2 b x / 2 a + b^{2} / 4 a^{2} + c / a = y / a + b^{2} / 4 a^{2}$ (adding $b^{2} / 4 a^{2}$ on both sides)

$[x + b / 2 a]^{2} + c / a = y / a + b^{2} / 4 a^{2}$

$[x + b / 2 a]^{2} = y / a + b^{2} / 4 a^{2} - c / a$

$x + b / 2 a = \pm \sqrt [y / a + b^{2} / 4 a^{2} - c / a]$

$x + b / 2 a = \pm \sqrt [4 a y + b^{2} - 4 a c] / \sqrt | 4 a^{2} |$

$x = - b / 2 a \pm \sqrt [4 a y + b^{2} - 4 a c] / \sqrt | 4 a^{2} |$