If n be a positive integer such that one of the roots of the quadratic equation  $4x^2-(4\sqrt{3}+4)x+\sqrt{3}n-24=0$ is an integer then the value of n is equal to____.

Let $\alpha$ be an integer satisfying the given equation, so that 
  $4{\alpha }^2-(4\sqrt{3}+4)\alpha +\sqrt{3n}-24=0\Rightarrow 4{\alpha }^2-4\alpha -24=\sqrt{3}(4\alpha -n).$
Since is irrational, it follows that  $4{\alpha }^2-4\alpha -24=0$  and  $4\alpha -n=0.$
Substituting  $\alpha =\frac{n}{4}$  into the first equation, we have
$4{\left(\frac{n}{4}\right)}^2-4=\left(\frac{n}{4}\right)-24=0$
$\begin{array}{l}\Rightarrow n^2-4n-96=0\\ \Rightarrow (n-12)(n+8)=0\\ \Rightarrow n=12,-8.\end{array}$   
Since n is a positive integer, we have $n=12.$