Applied Mathematics for Science and Engineering

2Algebraic Equations

Introduction

Algebraic equations are commonly encountered in science and engineering, and a few examples of where they originate include material balances for separation processes, force resolution in structures, flow in pipe networks, application of Kirchoff's rules to electric circuits and networks, radiative exchange in enclosures, solution of discretized differential equations, and balances in chemical equilibria. Though such problems are often thought of as elementary, cases can arise that offer greater challenge than the analyst might expect.

It is impossible to know exactly when an algebraic equation was solved for the first time, but there is evidence indicating that quadratic equations were solved by the Babylonians perhaps 3700 years ago. Heath (1964, reprinted from the 1910 Edition) notes that the “father” of algebra, Diophantus of Alexandria, authored Arithmetica in 13 books in the third century AD. Six of the original 13 books still exist, and Heath produced English translations of them in 1885 (with the second edition published in 1910). In the Arithmetica, Diophantus solves determinate equations of the first and second degree; for quadratic equations he sought only rational, positive solutions in either integral or fractional form. For example, he gives

(2.1)

and concludes that x = 78/325 or 6/25. Given an equation of the form ax² − bx + c = 0, ...