If the difference between the roots of the equation

Given: $x^{2}+a x+8=0$

Let $\alpha$ and $\beta$ are the roots of the equation.

Sum of the roots $=\alpha+\beta=\frac{-a}{1}=-a$.

Product of the roots $=\alpha \beta=\frac{8}{1}=8$

Given: $\alpha-\beta=2$

Then, $(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$

$\Rightarrow(\alpha+\beta)^{2}-2^{2}=4 \times 8$

$\Rightarrow(\alpha+\beta)^{2}-4=32$

$\Rightarrow(\alpha+\beta)^{2}=32+4=36$

$\Rightarrow(\alpha+\beta)=\pm 6$

$\alpha+\beta=-a=\pm 6$

$\therefore a=\pm 6$