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Q.
Suppose $k \in R$ and the quadratic equation $x^2-(k-3) x+k=0$ has at least one positive roots, then $k$ lies in the set:
Complex Numbers and Quadratic Equations
Solution:
Solution: Both the roots $\alpha, \beta$ will be non-positive if
$D \geq 0, \alpha+\beta \leq 0, \alpha \beta \geq 0 $
$\Rightarrow (k-3)^2-4 k \geq 0,(k-3) \leq 0, k \geq 0$
$\Rightarrow (k-1)(k-9) \geq 0, k \leq 3, k \geq 0 $
$\Rightarrow 0 \leq k \leq 1 $
Thus, quadratic equation will have at least one positive root if $k< 0$
or $k >1$ and $(k \leq 1$ or $k \geq 9)$
$\Rightarrow k \in(-\infty, 0) \cup[9, \infty)$