Page 36 - Standard Handbook Of Petroleum & Natural Gas Engineering

Algebra 25

and the roots

-b + db2 - 4ac
1.
2a
and

-b - .\/b2 - 4ac
2.
2a
The sum of the roots is -b/a and their product is c/a.
Third-degree equations (cubic equations), in the general case, have the form,
after division by the coefficient of the highest-order term,

xJ + ax2 + bx + c = 0

with the solution
x: = Ax, + B
where x, = x - a/3
A = 3(a/3)2 - b
B = -2(a/3)3 + b(a/3) - c
Exponential equations are of the form

ax = b
with the solution x = (log b)/(log a) and the root (log b)/(log a). The complete
logarithm must be taken, not only the mantissa.
Trigonometric equations are of the form
a cos x * b sin x

If an acute angle u is found, where
tan u = b/a

and an angle v (0' < v < 180') is found, where
cos2 v = c2/(az + b2)

the solution is x = f(u f v) and the roots are +(u + v) and +(u - v), depending
on the sign of b.

Solution of Systems of Simultaneous Equations

A set of simultaneous equations is a system of n equations in n unknowns. The
solutions (if any) are the sets of values for the unknowns which will satisfy all
the equations in the system.