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Consider the equation ${x^2} + \alpha x + \beta = 0$ having roots $\alpha ,\beta $ such that $\alpha \ne \beta $ .Also consider the inequality $\left| {\left| {y - \beta } \right| - \alpha } \right| < \alpha $ ,then
inequality is satisfied by exactly two integral values of $y$
inequality is satisfied by all values of $y \in (-4, 2)$
Roots of the equation are of same sign
${x^2} + \alpha x + \beta > 0\,\forall \,x \in \,\left[ { - 1,0} \right]$
Solution
$\alpha, \beta$ are roots of $\mathrm{x}^{2}+\alpha \mathrm{x}+\beta=0$ is
$\alpha+\beta=-\alpha$ …….$(1)$
and $\alpha \beta=\beta$ …….$(2)$
from $(1) $ and $(2) \alpha=1, \beta=-2$
Now $\| y+2|-1|<1$
$\Rightarrow-1<|\mathrm{y}+2|-1<1 \Rightarrow 0<|\mathrm{y}+2|<2$
$\Rightarrow-2<\mathrm{y}+2<2$ and $\mathrm{X} \neq-2$
$\Rightarrow \mathrm{y} \in(-4,0)-\{-2\}$
