The smallest positive integral value of a , for which all the roots...

Let $t = x^2$, so the equation becomes $t^2 - at + 9 = 0$.
For all four roots of the original equation to be real and distinct, both roots of this quadratic must be positive and distinct.
Discriminant $> 0$: $a^2 - 36 > 0 \Rightarrow |a| > 6$. Since $a > 0$, we need $a > 6$.
Product of roots $= 9 > 0$ and sum of roots $= a > 0$, so both roots are positive.
The smallest positive integer satisfying $a > 6$ is $a = 7$.