#QUADRATIC EQUATIONS FULL#

The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. The writing of the Chinese mathematician Yang Hui (1238-1298 AD) represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.īy 1545 Gerolamo Cardano compiled the works related to the quadratic equations. His solution was largely based on Al-Khwarizmi's work.

#QUADRATIC EQUATIONS FULL#

The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions, as well as irrational numbers as solutions. He also described the method of completing the square and recognized that the discriminant must be positive, which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/ c = y). ( Brahmasphutasiddhanta (Colebrook translation, 1817, page 346) To the absolute number multiplied by four times the square, add the square of the middle term the square root of the same, less the middle term, being divided by twice the square is the value. In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive. Pythagoras and Euclid used a strictly geometric approach, and found a general procedure to solve the quadratic equation.

Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula.Įuclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. In the Sulba Sutras in ancient India circa 8th century BC quadratic equations of the form ax2 = c and ax2 + bx = c were explored using geometric methods.

There is evidence pushing this back as far as the Ur III dynasty. Find x and y by inspection of the values in (1) and (4).The original pair of equations were solved as follows: Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve a pair of simultaneous equations of the form:

Quadratic equations