The complex conjugate roots exists A = -1 - 4i or A = -1 + 12i.
How to estimate complex conjugate roots?
If one of the roots exists w = B + 2i, then the other root exists its conjugate w = B - 2i. So we can factorize the quadratic to
[tex]z^2+4z+20+iz(A+1) = (z-(B+2i))(z-(B-2i))[/tex]
Expand the right side and collect all the coefficients.
[tex]z^2+(4+(A+1)i)z+20 = z^2-2Bz+B^2+4[/tex]
From the z and constant terms, we have
[tex]$\left \{ {{4+(A+1)i = -2B} \atop {20 = B^2+4}} \right.[/tex]
From the second equation, we get
[tex]B^2 = 16[/tex]
B = ± 4
Then 4+(A+1)i = ± 8
(A + 1)i = 4 or (A + 1)i = -12
Since [tex]$\frac{1}{i} = -i[/tex], we have
[tex]$\frac{-A+1}{i} =4[/tex] or [tex]$\frac{-A+1}{i} =-12[/tex]
A+1 = -4i or A+1 = 12i
A = -1-4i or A = -1+12i
Therefore, the complex conjugate roots exists A = -1-4i or A = -1+12i.
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