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Q.
If $(2-3 i)$ is a root of the equation $x^{3}-b x^{2}+25 x+d=0$ (where $b$ and $d$ are real and $i=\sqrt{-1}$ ), then value of $b$ is equal to
Complex Numbers and Quadratic Equations
Solution:
If one root is $2-3 i$, then other root is $2+3 i$
sum of roots $=b=\alpha+2-3 i+2+3 i$
$=4+\alpha$.....(i)
sum of pair of roots $=25$
$=(2-3 i)(2+3 i)+\alpha(2-3 i)+\alpha(2+3 i)=13+4 \alpha$
$=13+4 \alpha$
$\Rightarrow \alpha=3$ .....(ii)
Using (i) and (ii) $\Rightarrow b=7$