Quadratics and Cubics


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Quadratic equations were known and dealt with by Babylonians, as early as 1800BC; in the Sulba Sutras (sutra text relating to altar construction in Hinduism and the main source of our understanding of Indian mathematics from the Vedic period - 1500-500BC) the quadratics of the form

        and                were explored using geometric methods.

Al-Khwarizmi also, in relation to the technique of 'completing the square', worked with quadratic equations.

The quadratic formula for solving equations can be easily derived from the completed square form of a quadratic.

When we say that Babylonians for example, or Egyptians, knew about quadratic or cubic equations and how to solve them, we mean that they used certain prescribed methods to solve problems that we would today write, using algebra, as such equations.

Euclid in his Elements (c. 300) developed a geometrical approach to solving quadratics in Proposition 11, Book II. Propositions VI-28 and VI-29 considered equations such as


where a and b represented the lengths of line segments.

The great advancements to the solutions of quadratic and cubic equations came through two Italian mathematicians from the 15 th century: Luca Pacioli (1445-1517) and Scipione del Ferro (1465-1526). Whilst Luca published his results in one of the famous books on mathematic Summa de arithmetica, geometria, proportioni et proportionalita (Venice 1494), in which he wrote pretty much all that was known of mathematics in Italy at the time, del Ferro left his work unpublished in a manuscript which was later adopted by some of his friends who were interested in mathematics.




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