Quadratic Equations
A quadratic equation is a second order, univariate polynomial with constant coefficients and can usually be written in the form: ax2 + bx + c = 0, where a 0. In about 400 B.C. the Babylonians developed an algorithmic approach to solving problems that give rise to a quadratic equation. This method is based on the method of completing the square. Quadratic equations, or polynomials of second-degree, have two roots that are given by the quadratic formula: x = (-b +/- (b2 - 4ac))/2a. There is another form of this equation yielding the roots for a quadratic equation that is obtained by first dividing the original quadratic equation through by x: x = (2c)/(-b +/- (b2 - 4ac)). This equation, which provides the roots to the quadratic equation, is often useful when b2 > 4ac. In these cases the usual form providing roots to the quadratic equation can yield erroneous numerical results.
The earliest solutions to quadratic equations involving an unknown are found in Babylonian mathematical texts that date back to about 2000 B.C.. At this time the Babylonians did not recognize negative or complex roots because all quadratic equations were employed in problems that had positive answers such as length. The theory involving quadratic equations, and all polynomial equations, was flawed prior to the 17th century because of this idea. In 400 B.C. the Babylonians developed the quadratic formula used to find the roots of quadratic equations. About 100 years later Euclid formulated a geometrical approach to solving quadratic equations. His approach involved determining a length that would be the root of a quadratic equation. There were many other methods used in ancient times to determine the roots to quadratic equations. The Egyptians employed the false position method which involved approximating x to make part of the quadratic equation easy to calculate. Then a scaling factor was incorporated to find the root of the original equation.
Greek mathematicians employed the iteration method in which a positive root of a quadratic equation is approximated and substituted for the unknown. Then this is used to form another approximation hat is substituted and calculated. This process is repeated until the real root is determined. Between 598 and 665 A.D. Brahmagupta, an Indian mathematician, advanced the Babylonian methods to almost modern methods. Indian and Chinese mathematicians recognized negative roots to quadratic equations. Al-Khwarizmi, an Arab mathematician, developed a classification of quadratic equations in the 9th century. They were classified into one of six different types depending on which coefficients were negative. He wrote six chapters with each chapter devoted to a different type of equation. The equations were composed to three types of quantities: roots, squares of roots and numbers, and numbers. In each chapter al-Khwarizmi described the rule used for solving each type of quadratic equation and then presented a proof for each example. Later, in 1145, Abraham bar Hiyya Ha-Nasi, also known by the Latin name Savasorda, published a book that was the first to give the complete solution of the quadratic equation. Over the next few hundred years several mathematicians advanced the study of quadratic and cubic equations. Near the end of the 18th century Carl Friedrich Gauss, a German mathematician, gave a proof that showed every polynomial equation has at least one root. The root may not be able to be expressed as an algebraic formula involving the coefficients of the equation but a root did exist. Eventually a team of three international mathematicians combined and showed that only polynomials of degree five or less could be solved via a general algebraic formula. It is this set of polynomials that the theory of equations focuses on.
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