The Proof of Sridharacharya’s Quadratic Formula: An Insight into Mathematical History

Introduction to the Quadratic Equation and Sridharacharya

The Sridharacharya formula, more commonly known as the quadratic formula, is a fundamental tool in algebra used to find the roots of a quadratic equation. A quadratic equation is an equation of the form:

(ax^2 bx c 0)

where (a, b,) and (c) are constants and (a eq 0). The quadratic formula is:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

Historical Context

The proof of the quadratic formula has been a subject of mathematical inquiry for centuries. As noted by Bertrand Russell in A History of Western Philosophy, the rigor of mathematics can withstand the test of time. Sridharacharya's formulation of this formula in the 9th century is celebrated for its elegance and utility. However, it is fascinating to consider that methods to solve quadratic equations existed as early as 2000 BC on Babylonian clay tablets. This historical perspective underscores the timeless nature of mathematical truths.

Deriving the Quadratic Formula

To derive the quadratic formula, we start with the standard form of the quadratic equation:

(ax^2 bx c 0)

We divide the entire equation by (a,) assuming (a eq 0,) to get:

(x^2 frac{b}{a}x frac{c}{a} 0)

Next, we rearrange the equation:

(x^2 frac{b}{a}x -frac{c}{a})

To complete the square, we take half of the coefficient of (x,) which is (frac{b}{a},) square it, and add it to both sides:

(x^2 frac{b}{a}x left(frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2)

This simplifies to:

(left(x frac{b}{2a}right)^2 -frac{c}{a} frac{b^2}{4a^2})

Combining the right-hand side under a common denominator, we get:

(left(x frac{b}{2a}right)^2 frac{b^2 - 4ac}{4a^2})

Taking the square root of both sides, we obtain:

(x frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a})

Isolating (x,) we find:

(x -frac{b}{2a} pm frac{sqrt{b^2 - 4ac}}{2a})

This simplifies to:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

Therefore, we arrive at the quadratic formula:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

This formula provides the roots of the quadratic equation (ax^2 bx c 0) and is widely used in mathematics for solving such equations.

Conclusion

The proof of the quadratic formula is not only a testament to the power of algebraic manipulation but also a reflection of the enduring nature of mathematical knowledge. From Babylonian clay tablets to the work of Sridharacharya, the techniques for solving quadratic equations have evolved but the core formula remains unchanged. This enduring utility is a reminder of the beauty and complexity of mathematics, as eloquently captured by Bertrand Russell.