Problem:
Let i=−1. The product of the real parts of the roots of z2−z=5−5i is
Answer Choices:
A. −25
B. −6
C. −5
D. 41
E. 25
Solution:
The quadratic formula leads to the roots
z=21(1±21−20i)
To find 21−20i, let (a+bi)2=21−20i where a and b are real. Equating real and imaginary parts leads to a2−b2=21 and 2ab=−20. Solve these equations simultaneously:
The product of the real parts of these two roots is −6.
Note. One could also use the equation a2−b2=21 together with a2+b2=∣a+bi∣2=∣21−20i∣=29 and solve simultaneously to obtain 2a2=50, from which it follows that a=±5.