Question

Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic z^2 + (a + 8i) z + (-39 + bi) = 0. What is the other root?

I tried working this problem and so far I got that (a+8i) - (4-5i)= (-39-bi)/(4-5i). Im not sure how to solve this problem.

Answer 1

I'd just multiply:

(4-5i)^2 + (a+8i)(4-5i) + (-39+bi) = 0+0i
Expand all that out and you have two equations in a and b. I get

a = 2
b = 18

So, that leaves us with
z^2 + (2 + 8i) z + (-39 + 18i) = 0
and the other root is thus

-6 -3i