If the equation ${\mathrm{x}}^2+px+\mathrm{q}=0 \mathrm{and} {\mathrm{x}}^2+\mathrm{qx}+\mathrm{p}=0$, have a common root, then $\mathrm{p}+\mathrm{q}+1=$

Let $\mathrm{\alpha }$ is the common root, so ${\mathrm{\alpha }}^2+\mathrm{p\alpha }+\mathrm{q}=0 ...\left(\mathrm{i}\right)$

and ${\mathrm{\alpha }}^2+\mathrm{q\alpha }+\mathrm{p}=0 ...\left(\mathrm{ii}\right)$

from $\left(\mathrm{i}\right)-\left(\mathrm{ii}\right)$,

$(a-b)\mathrm{\alpha }+20=0$

$\Rightarrow (\mathrm{p}-\mathrm{q})\mathrm{\alpha }+(\mathrm{q}-\mathrm{p})=0\Rightarrow \mathrm{\alpha }=1$

Put the value of $\alpha$ in (i), $\mathrm{p}+\mathrm{q}+1=0.$