#### Quadratic Equations – Explore Trans -Disciplinary Learning

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##### Indian Mathematicians

Indian mathematicians such as Brahmagupta (7th century) and later Bhaskara II (12th century) made significant advancements. Brahmagupta provided solutions to **Quadratic Equations** in the form ax2+bx=c, and Bhaskara developed methods for solving these equations more systematically. Example: Bhaskara’s approach would solve an equation like x2+12x=36 by using techniques that closely resemble our modern methods of completing the square and using the quadratic formula.

##### Ancient Babylonian Methods

The study of quadratic equations dates back to ancient civilizations. The Babylonians, around 2000 BC, were among the first to solve quadratic equations. Their approach was geometric rather than algebraic. They used tables and methods to solve problems that we would now describe using **Quadratic Equations** . Their methods often involved a form of completing the square, which is remarkably similar to techniques used today. Example: Suppose a Babylonian problem was to find the side length xxx of a square that, when increased by 10 and the area, equals 39. Mathematically, this can be written as: x2+10x=39 The Babylonians would rearrange and complete the square, eventually finding the solution through their tabular methods. www.thegreenschoolbangalore.com Greek

###### Greek Contributions

The Greeks, particularly through the work of mathematicians like Euclid, also made significant contributions. They used geometric methods to solve **Quadratic Equations** and explored the relationships between different shapes. Euclid’s “Elements” include geometric constructions that can solve quadratic problems, furthering the development of mathematical techniques. Example: A Greek problem might involve finding the length of the side of a square given its area, which would involve solving x2=A. While straightforward, it laid the groundwork for more complex problems.

##### Islamic Golden Age

During the Islamic Golden Age, mathematicians like Al-Khwarizmi (9th century) significantly advanced the study of **Quadratic Equations** Al-Khwarizmi’s work “Al-Kitab al-Mukhtasar fi Hisab al-Jabr walMuqabala” (The Compendious Book on Calculation by Completion and Balancing) outlined systematic methods for solving quadratic equations. His work introduced the concepts of “reduction” and “balancing” to move terms across the equation, which is fundamental to modern algebra. Example: Al-Khwarizmi classified quadratic equations into several standard forms and solved them using geometric methods. For instance, to solve x2+10x=39, he would geometrically complete the square to find x=3

##### Renaissance Europe

In Renaissance Europe, mathematicians such as François Viète (16th century) and later René Descartes (17th century) began to formalize algebraic notation and methods. Descartes’ work laid the foundations for analytic geometry, linking algebra and geometry in new ways. Example: Viète introduced a notation system that allowed for more abstract manipulation of equations, setting the stage for modern algebra. Descartes’ work allowed to be graphed as parabolas, bridging the gap between algebra and geometry