36 CHAPTER 1 • COMPLEX NUMBERS
yFigure 1.20 The point z =Bi and its three cube roots, zo, zi. and z2.ls the quadratic formula valid in the complex domain? The answer is yes,
provided we are careful with our terms.
• EXAMPLE 1.21 Find all solutions to the equation z^2 + (1 + i) z + 5i = O.
Solution The quadratic formula gives
l- c1 +i) + [c1 +i)
2
-4(1)(5i)r -c1 + i) + (- 18i)t
z = 2(1) = 2
As -18i = 18 e•(- ~), Equations (1- 45 ) give (- 18i) ~ = 1st e/-~;'"w), fork= 0
and 1. In Cartesian form, this expression reduces to 3 - 3i and - 3 + 3i. Thus,
our solution set is {-(l+i)+ ( 2 3 -^3 il , -(l+i)+ ( - 2 3+^3 il} ' or {1 - 2i , - 2 + i}.