If $d=\mathrm{degree}\left(p\,x\right)$ and $a=\mathrm{lcoeff}\left(p\,x\right)$ then the discriminant is

$p≔a{x}^2+bx+c$

$p≔a{x}^2+bx+c$

$\mathrm{discrim}\left(p\,x\right)$

$-4ac+b^2$

$l≔5x^2+x-1$

$l≔5x^2+x-1$

$\mathrm{discrim}\left(l\,x\right)$

$21$

$\mathrm{solve}\left(l\,x\right)$

$-\frac{1}{10}+\frac{\sqrt{21}}{10},-\frac{1}{10}-\frac{\sqrt{21}}{10}$

$k≔\frac{x^2}{2}-x+3$

$k≔\frac{1}{2}{x}^2-x+3$

$\mathrm{discrim}\left(k\,x\right)$

$\mathrm{-5}$

$\mathrm{solve}\left(k\,x\right)$

$1+\mathrm{I}\sqrt{5},1-\mathrm{I}\sqrt{5}$