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Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^{2}+\alpha z+\beta=0$ has two distinct roots on the
line $\operatorname{Re} z=1$, then it is necessary that

1) $|\beta|=1$
2) $\beta \in(1, \infty)$
3) $\beta \in(0,1)$
4) $\beta \in(-1,0)$
in Mathematics by Diamond (74,866 points) | 9 views

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Since $\alpha, \beta$ are real, Real part of the different roots is 1, sum...and product of the roots are real $\Rightarrow$ The roots may be $1+a i$ and $1-a i, a \in R^{*}$ $\Rightarrow \beta=1+a^{2}>1 \quad \Rightarrow \beta \in(1, \infty)$

by Diamond (74,866 points)

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