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Q.
If the quadratic equation $z^{2}+(a+i b) z+c+i d=0$, where $a, b, c, d$ are non-zero real numbers, has a real root, then
Complex Numbers and Quadratic Equations
Solution:
Let $\alpha$ be a real root of the given equation.
Then, $\alpha^{2}+(a+i b) \alpha+c+i d=0$
$\Rightarrow \alpha^{2}+a \alpha+c=0 $ and $ b \alpha+d=0$
$\Rightarrow \left(-\frac{d}{b}\right)^{2}+a\left(-\frac{b}{d}\right)+c=0$
$\Rightarrow \frac{d^{2}}{b^{2}}-\frac{a d}{b}+c=0$
or $ d^{2}-a b d+b^{2} c=0$