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If $$\alpha, \beta \in R$$ 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:
Since $$\alpha, \beta \in \mathbb{R}$$, the polynomial $$z^2 + \alpha z + \beta = 0$$ has real coefficients. Because $$1 - 2i$$ is a root, its complex conjugate $$1 + 2i$$ must also be a root.
The sum of the roots gives $$-\alpha = (1 - 2i) + (1 + 2i) = 2$$, so $$\alpha = -2$$.
The product of the roots gives $$\beta = (1 - 2i)(1 + 2i) = 1 + 4 = 5$$.
Therefore, $$\alpha - \beta = -2 - 5 = -7$$.
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