This website will be removed on December 31st, 2017.
If you are the site owner, please visit the Server Decommissioning page for more details.

Hindu Solution

The Hindus found a solution to the quadratic ax2 + bx = c. It is essentially our present method of "completing the square." The Hindus realized that a quadratic with real roots has two roots, but they didn't always find both. They discarded negative and imaginary roots, calling the former "inadequate" and said about the latter, "as in the nature of things, a negative is not a square, it has therefore no square root" (Mahavira).

Below is Brahmagupta's rule for finding one of the two positive roots of the quadratic equation x2 - 10x = -9. The first column is the solution as translated in D.E. Smith's History of Mathematics, Vol. II, 1925, p.445. Note that a dot above a number symbolizes the opposite or negative of that number. The second column is the solution in modern notation, and the third is a generalization for ax2 + bx = c.

Hindu Quadratic Solution

(NCTM, Historical Topics in Algebra, 1971, p. 44.)

The last equation should look familiar. It is a version of the quadratic formula. It is one of the roots from the quadratic formula, and in this case there is a plus sign under the radical instead of a minus. This is because the formula is set up for the equation ax2 + bx = c (or ax2 + bx - c = 0) instead of ax2 + bx + c = 0, which is used today.