It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic equations. This is an over simplification, for the Babylonians had no notion of 'equation'. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation. The method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.
In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities.
Hindu mathematicians took the Babylonian methods further so that Brahmagupta (598-665 AD) gives an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a color was used, and sometimes several different unknowns occur in a single problem.
The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each). The different types arise since al-Khwarizmi had no zero or negatives. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. x, x2 and numbers.
- Squares equal to roots.
- Squares equal to numbers.
- Roots equal to numbers.
- Squares and roots equal to numbers, e.g. x2 + 10x = 39.
- Squares and numbers equal to roots, e.g. x2 + 21 = 10x.
- Roots and numbers equal to squares, e.g. 3x + 4 = x2.
Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.
Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.
A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra Luca appearing in small print but not on the title page. In many ways the book is more a summary of knowledge at the time and makes no major advances. The notation and setting out of calculations is almost modern in style:
Quadratics are equations of the second degree, having the form ax^2+bx+c=0, for a,b,c constant. They, or their equivalents, have been around for at least 4,000 years. The solution, which we know today as the "quadratic formula," has been around for a few hundred years in the exact form we use today but took thousands of years to formulate. Impediments included formulations that were largely geometric instead of numerical and unwillingness to recognize a second root, irrational numbers, or even negatives.
Quadratics are equations of the second degree, having the form ax^2+bx+c=0, for a,b,c constant. They, or their equivalents, have been around for at least 4,000 years. The solution, which we know today as the "quadratic formula," has been around for a few hundred years in the exact form we use today but took thousands of years to formulate. Impediments included formulations that were largely geometric instead of numerical and unwillingness to recognize a second root, irrational numbers, or even negatives.