If α , β ∈ R are such that 1 - 2i

Question:

If $\alpha, \beta \in R$ are such that $1-2 i$ (here $i^{2}=-1$ ) is a root of $z^{2}+\alpha z+\beta=0$, then $(\alpha-\beta)$ is equal to :

  1. $-3$

  2. -7

  3. 7

  4. 3


Correct Option: , 2

Solution:

$\because \alpha, \beta \in \mathrm{R} \Rightarrow$ other root is $1+2 \mathrm{i}$

$\alpha=-($ sum of roots $)=-(1-2 \mathrm{i}+1+2 \mathrm{i})=-2$

$\beta=$ product of roots $=(1-2 \mathrm{i})(1+2 \mathrm{i})=5$

$\therefore \alpha-\beta=-7$

option (2)

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