Question

Find all values of p or which the roots of the equation $(p-3)x^2-2px+5p+0$ and positive

Answer 1

$(p-3)x^2-2px+5p=0$

For roots to be real, $D\ge 0$

$(-2p{)}^2-4(5p\left)\right(p-3)\ge 0$

$\Rightarrow 4p^2-4(5p^2-15p)\ge 0$

$\Rightarrow p^2-5p^2+15p\ge 0$

$\Rightarrow -4p^2+15p\ge 0$

$\Rightarrow 4p^2-15p\le 0$

$\Rightarrow p(4p-15)\le 0$

$\Rightarrow 0\le b\le 15/4$

For roots to be $+$ve, $p\left(0\right)>0$

$\Rightarrow 5p\ge 0$

& vertex should be $>0$

$\Rightarrow \frac{-b}{2a}>0$

$\Rightarrow \frac{2p}{2(p-3)}>0$

$\Rightarrow \frac{p}{p-3}>0$

$\Rightarrow p<0$ or $p>3$

$\Rightarrow p\in (3,15/4)$.