Problem:
Suppose that x,y, and z are complex numbers such that xy=−80−320i, yz=60, and zx=−96+24i, where i=−1. Then there are real numbers a and b such that x+y+z=a+bi. Find a2+b2.
Thus xyz=±240(5+3i). Dividing this equation by each of the three given equations yields x=±(20+12i),y=±(−10−10i), and z=±(−3+3i). Hence x+y+z=±(7+5i) and (a,b)=±(7,5). Thus a2+b2=72+52=74.