QUADRATIC EQUATIONS 5
“Babylonian Treatment of ”
A quadratic refers to ax2 + bx + c,a, b and c act as given figures,where a ≠0. A quadratic equation’snormal form is ax2 + bx + c =0 (AMSI,2011). The equation aims at finding values that make the formfactual, or demonstrate the absence of such values. Thus, quadraticsolutions can have a solution, lack a solution or have more than onesolution. Three fundamental techniques are applicable when solvingquadratic equations. These are factoring, using the quadratic formulaand completing a square (AMSI,2011). Quadratic equations have been in existence from 2000BC, whenBabylonians used them.
The Babylonians knowledge about the quadratic equations differs fromthe form currently in use today. This is because they were unaware ofthe idea of a formula. In place of a formula, the Babyloniansemployed numerical procedures. For instance, supposing that theywanted to find a solution to the quadratic equationthesolution would be provided depending on the equations coefficients.The coefficient one acts as the coefficient of square X, then addedto X’s coefficient. Today, the numerical procedure by theBabylonians resembles the quadratic equation solution.
It is however, important to note that the Babylonians did not providea solution that comprised a negative square root. When currentlyemploying the quadratic formula to find a solution to the equation,the answer must have a plus or negative of the square root. Thisdiffers from the Babylonians who only applied the plus sign. TheBabylonians, when using quadratic equations, wrote the equation in astructure where all figures were plus, with the main coefficientbeing one. This resulted in four forms of numerical procedures. Theseare
In addition, the Babylonians had numerical formulas for two of thequadratic equation forms.
It is uncertain if the Babylonians had formulas for the remaining twoforms. However, they did have methods for finding solutions to theforms. The method is best demonstrated using an illustration. Whensolving x squared added to b, which equals ax, with a and b beingpositive figures, the Babylonians would start by introducing variabley. They then add x to y, which adds up to the coefficient of x aswell as x multiplied by y equaling to an invariable term.
Babylonian solving technique
This is set as X + Y = a while XY = b
The motivation behind the solving technique derives from theBabylonians’ curiosity to solve problems when given a rectanglemeasuring X by Y. The rectangle would have semi-perimeter (a), inaddition to an area (b). The Babylonians had to determine value X andY. Supposing one finds a solution to the rectangle it starts bysubstituting (y) into (xy), which equals (b). The outcome is aquadratic equation. Hence, the Babylonian solving technique withknowledge of the semi perimeter as well as its area is alike tosolving a quadratic equation.
The Babylonians solved the quadratic equation by introducing adifferent variable, z. They then set X to add to X’s halfcoefficient. Hence, when X is added to Z plus Y it equals to half thecoefficient, while subtracting Z. This means that through adding X toY, (a), becomes the solution as demonstrated above. The first set (x+ y = a), is no longer important. What is needed is the set of XY =b. This results in a straightforward solution to Whensolving usingthe new variable Z, the Babylonians used aswell as xy = b, X is set as X equals (a) divided by 2 added by Z. Atthe same time (y) equals (a) divided by 2 subtract (z). Such asetting means that (x + y = a).
It is apparent that the Babylonians understanding of mathematicsdiffers from what we currently use. The Babylonians seem to havefocused more on finding answers, and placed minimal emphasis on themethod used in solving an equation (AMSI,2011). Mathematicians argue that the significance of studyingmathematics derives from becoming aware of how to reason via aprocess, contrary to getting answers to isolated submissions.
Based on the analysis of Babylonians approach to quadratic equations,I conclude that their technique of solution lacks evidence. TheBabylonian technique is articulated as a sequence of steps. Inaddition, they found solutions to non-linear concurrent equations,which resulted in set algebra to quadratics. For instance, (x + y =10, or xy = 5). Although their technique is interesting, it is notapplicable in today’s classroom. The reason is explained using anexample x2 −2x −8 = 0.
The first step involves taking the constant from the other side andfactoring it to the left, x(x −2) = 8. Then (a) is placed as the mean of the terms, a = x-1.Hence, x = a + 1, (x – 2) = a – 1. The final step is substitutingas well as solving.
x (x −2) = 8, (a +1)(a −1) = 8, a2 −1 = 8, a2 =9. Since negative figures were not used by Babylonians, the solutioncan only be (a = 3, hence x = 4). This is an irrational solution,which explains why the method is not applicable in the classroom.
Australian Mathematical Sciences Institute, AMSI. (2011). Theimproving mathematics education in schools: Quadratic equations.Retrieved from http://www.amsi.org.au/teacher_modules/Quadratic_Equations.html#The_Babylonian_m ethod_of_solution