If , R are such that 1-2i (here i^2=-1) is a root of z^2+ z+=0, then (-) is equal to

Question

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

Accepted Answer

Since coefficients are real, the other root is $1+2i$. Sum of roots $=-\alpha=2 \implies \alpha=-2$. Product of roots $=\beta=(1-2i)(1+2i)=5$.

$$\alpha-\beta=-2-5=-7.$$

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