Page 526 - Modelling in Transport Phenomena A Conceptual Approach

506 APPENDIX A. MATHEMATICAL PRJ3LIMINARIES

A.7 THE ROOT OF AN EQUATION

In engineering problems, we frequently encounter with equations of the form
f (XI = 0 (A.7-1)

and want to determine the values of z satisfying Eq. (A.7-1). These values are
called the roots of f(z) and may be real or imaginary. Since imaginary roots appear
as complex conjugates, the number of imaginary roots must always be even.
The function f(z) may be a polynomial in z or, it may be a transcendental
equation involving trigonometric and/or logarithmic terms.

A.7.1 Roots of a Polynomial

If f(z) is a polynomial, then Descartes’ rule of sign determines the maximum
number of real roots:

0 The maximum number of real positive roots is equal to the number of sign
changes in f(z) = 0.

0 The maximum number of real negative roots is equal to the number of sign
changes in f(- z) = 0.

In applying the sign rule, zero coefficients are regarded as positive.

A.7.1.1 Quadratic equation

The roots of a quadratic equation

ax2 + bz+c= 0 (A.7-2)

are given as
-bfi/m
z1,2 = (A.7-3)
2a
If a, b, c are real and if A = b2 - 4ac is the discriminant, then

0 A > 0; the roots are real and unequal,

0 A = 0; the roots are real and equal,
0 A < 0; the roots are complex conjugate.