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Q.
Minimum possible number of positive roots of the equation $x^2-(\lambda+1) x+(\lambda-2)=0$ where $\lambda \in R$ is
Complex Numbers and Quadratic Equations
Solution:
$D =(\lambda+1)^2-4(\lambda-2) $
$D =\lambda^2-2 \lambda+9=(\lambda-1)^2+8>0 $
Hence roots are distinct
Case-1: $$ If $\lambda<2$, then $f (0)<0 \Rightarrow 1$ root positive and 1 root negative
Case-2: If $\lambda>2$, then $f(0)>0$ and $\frac{-b}{2 a}>0$. Hence both roots are positive and distinct.
Case-3: If $\lambda=2$, equation is $x^2-3 x=0 \Rightarrow x=0,3$
$\therefore $ Minimum number of positive roots of equation is 1.$