Complex Roots of a Quadratic Equation

Date: 10/25/1999 at 12:01:05
From: Aoun Family
Subject: Complex Numbers

Hi,

I have a question concerning complex numbers:

The roots of the quadratic equation z^2+pz+q = 0 are 1+i and 4+3i. 
Find the complex numbers p and q. This was easy, the answers were:

     p = -5 - 4i   and   q = 1 + 7i

Now for the second part of the question. It is given that 1+i is also 
a root of the equation

     z^2 + (a+2i)z + 5+ib = 0

where a and b are real. Determine the values of a and b. I am getting 
a and b as complex numbers.

Can you help me?
Jad Aoun

Date: 10/25/1999 at 17:44:23
From: Doctor Schwa
Subject: Re: Complex Numbers

Jad,

Very nicely done on the first problem.

For the second problem, there are certainly answers where a and b are 
complex, but there is an answer where they are both real, too. If 
you're assuming that (1+i) is a root, then you can plug in z = 1 + i, 
and the equation should be true. Since it's a complex number equation, 
the real and imaginary parts must both equal 0, so you get two 
equations with two (real) unknowns, a and b.

If that hint doesn't make sense to you, please write back and I'll try 
to explain it in a different way that might work better for you.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/