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  4. The range of a such that the quadratic equation x2+(a-3) x+a=0 has two distinct positive roots, is

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Q. The range of a such that the quadratic equation $x^2+(a-3) x+a=0$ has two distinct positive roots, is

Complex Numbers and Quadratic Equations

Solution:

$D>0 \Rightarrow (a-3)^2-4 a>0 \Rightarrow a^2-10 a+9>0 \Rightarrow(a-9)(a-1)>0 \ldots \ldots . .(1)$
$\frac{-b}{2 a}>0 \Rightarrow \frac{3-a}{2}>0 \Rightarrow a<3$ .....(2)
and product of the roots $>0 \Rightarrow a>0 \ldots . .$. (3)
$(1),(2)$ and $(3) \Rightarrow a \in(0,1)$