Question

Find all possible values of 'a' for which exactly one root of the quadratic equation $x^2-(a+1)x+2a=0$ lie in the interval $(0,3)$

Answer 1

$x^2-\left(a+1\right)x+29=0$ $1\left(0,3\right)$ $p=0,q=3$

For exactly one root

$f\left(p\right).f\left(a\right)<0$

$\Rightarrow f\left(a\right)\times f\left(3\right)<0$

$\Rightarrow 2a\times \left[9-{\left(a+1\right)}^3+2a\right]<0$

$\Rightarrow 2a\left(9-3a-3+2a\right)<0$

$\Rightarrow 2a\left(6-a\right)<0$

$a\in \left(-\infty ,0\right)\cup \left(6,\infty \right)$