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?
+36916 Isn't it just 4 + 5i ? Complex roots come in pairs like g +- k i
After a few calulations I find the other root to be
-6-3i
+118608 Thanks EP.
I want to give it a go myself. 
z^2 + (a + 8i) z + (-39 + bi) = 0.
\([z-(4-5i)][z-(p+qi)]=z^2 + (a + 8i) z + (-39 + bi) \\ \text{Equating coefficients}\\ -(4-5i)-(p+qi)=a+8i\qquad and \qquad (4-5i)(p+qi)=-39+bi\\ -4+5i-p-qi=a+8i\qquad \qquad and \qquad 4p+5q+4qi-5pi=-39+bi\\ -4-p+(5-q)i=a+8i\qquad \qquad and \qquad 4p+5q+(4q-5p)i=-39+bi\\ -4-p=a \qquad 5-q=8\qquad \qquad and \qquad 4p+5q=-39 \qquad 4q-5p=b \\ -4-p=a \qquad \color{red}q=-3 \color{black}\qquad \qquad and \qquad 4p+5q=-39 \qquad 4q-5p=b \\ -4-p=a \qquad \qquad \qquad \qquad and \qquad 4p-15=-39 \qquad -12-5p=b \\ -4-p=a \qquad \qquad \qquad \qquad and \qquad \qquad \color{red}p=-6 \color{black}\qquad -12-5p=b \\ -4+6=a \qquad \qquad \qquad \qquad and \qquad \qquad \qquad \quad \color{black}\qquad -12+30=b \\ \color{red}a=2 \qquad \qquad\qquad \qquad \qquad and \qquad \qquad \qquad \quad \qquad\qquad b=18 \\\)
So we have
\([z-(4-5i)][z-(-6-3i)]=z^2 + (2 + 8i) z + (-39 + 18i) \\\)
I have not rechecked by expanding. But I also get the other root to be -6-3i.
Is there an easier way to do it?
Coding:
[z-(4-5i)][z-(p+qi)]=z^2 + (a + 8i) z + (-39 + bi) \\
\text{Equating coefficients}\\
-(4-5i)-(p+qi)=a+8i\qquad and \qquad (4-5i)(p+qi)=-39+bi\\
-4+5i-p-qi=a+8i\qquad \qquad and \qquad 4p+5q+4qi-5pi=-39+bi\\
-4-p+(5-q)i=a+8i\qquad \qquad and \qquad 4p+5q+(4q-5p)i=-39+bi\\
-4-p=a \qquad 5-q=8\qquad \qquad and \qquad 4p+5q=-39 \qquad 4q-5p=b \\
-4-p=a \qquad \color{red}q=-3 \color{black}\qquad \qquad and \qquad 4p+5q=-39 \qquad 4q-5p=b \\
-4-p=a \qquad \qquad \qquad \qquad and \qquad 4p-15=-39 \qquad -12-5p=b \\
-4-p=a \qquad \qquad \qquad \qquad and \qquad \qquad \color{red}p=-6 \color{black}\qquad -12-5p=b \\
-4+6=a \qquad \qquad \qquad \qquad and \qquad \qquad \qquad \quad \color{black}\qquad -12+30=b \\
\color{red}a=2 \qquad \qquad\qquad \qquad \qquad and \qquad \qquad \qquad \quad \qquad\qquad b=18 \\
+36916 Hey Melody!
That is what I did (on paper) to find the other root.....I got the same numbers as you did for a and b. I just didn't feel like transcribing it all on the keyboard when I was done ! 
(Maybe for a REGISTERED user who wanted derivation I would have...)
The question states a and b are real.....well , yes they are, but 'b' is part of b i (which obviously is not real)...a bit misleading, but at least we both figured it out....
