A NOVEL CONCEPT: THE SOLUTION OF ANY QUADRATIC EQUATION IN ONE UNKNOWN QUANTITY IN REAL NUMBERS - Prabir Chandra Bhattacharyya (Journal of Mechanics of Continua and Mathematical Sciences, 2025)
Conventionally the solution of any quadratic equation in one unknown quantity (say x) is represented in real or complex numbers. Instead of finding the solution of any quadratic equation in one unknown in complex numbers, the author introduced Bhattacharyya's Theorem – 1 & 2 to find its solution in real numbers only. Bhattacharyya's Theorem – 1 states that the square root of any negatively directed number is a negatively directed number and Bhattacharyya's Theorem -2 states that the square of any negatively directed number is a negatively directed number. Both theorems are based on the newly invented concept of the Theory of Dynamics of Numbers by the author. To find the root of any quadratic equation it must satisfy the two criteria: 1) The value of x must satisfy the equation (as conventional method). (2) The inherent nature of x must satisfy the quadratic equation (new concept). The inherent nature of x may be a positively directed number, a negatively directed number, or a neutral number which can be determined depending on the constant term, c<0, c>0, or c= 0 respectively of the quadratic equation. The author states that the quadratic expression which is factorizable into two linear functions may be defined as a pseudo-quadratic equation but all factorizable quadratic equations are not pseudo-quadratic equations. Using the unique concept of the Theory of Dynamics of Numbers the solution of the quadratic equation, ax2+bx+c=0, in one unknown quantity (say x) can be determined in real numbers only even if the discriminant, b2 - 4ac < 0, without using the concept of the complex numbers.