If the difference of the roots of the equation x

For $x^{2}-P x+8=0$

Let us suppose two roots are $z_{1}$ and $z_{2}$

given : $z_{1}-z_{2}=2$

also $z_{1} z_{2}=8$n (product of roots)

i. e. $\left(z_{2}+2\right) Z_{2}=8$

i. e. $z_{2}^{2}+2 z_{2}=8$

i.e. $z_{2}^{2}+2 z_{2}-8=0$

i. e. $z_{2}^{2}+4 z_{2}-2 z_{2}-8=0$

i. e. $\left(z_{2}+4\right)\left(z_{2}-2\right)=0$

i.e. $z_{2}=-4$ or $z_{2}=2$

$\Rightarrow z_{1}=-2$ or $z_{2}=4$

Since $P=z_{1}+z_{2}$

$\Rightarrow P=-4-2$ or $P=4+2$

i.e. $P=-6$ or 6