Appearances are Deceptive!

We have, $6x^{2}-77[x]+147=0$

$\Rightarrow\: [x]= \frac{6x^{2}+147}{77}$

$\Rightarrow\:[x]=\frac{6}{77}x^{2}+1.9$

$\Rightarrow\:[x]> 1.9$

$\Rightarrow\:[x]=2,3,4,5,...$

Now, putting the values of [x] in $x^{2}=\frac{77[x]-147}{6}$ , it can be checked that the equation is satisfied only for [x]=3,8,9,10. Further, for, $[x]\geq11$ , the RHS of the equation becomes much larger than the max. value of LHS; hence, we don't need to check for the values of [x] $\geq11$ .

So, $\alpha=4$ .

Hence, $7\alpha-13.8 = \boxed{14.2}$