The equation  $2{\left({\log }_{3}x\right)}^2-\left|{\log }_{3}x\right|+a=0$  has exactly four real solutions if  $a\in (0,\frac{1}{K})$ , then the value of K is __

on putting   ${\log }_{3}x=t,$  we get
  $2t^2-\left|t\right|+a=0$     …(i)
If  $t>0,$   then   $2t^2-t+a=0$   …(ii)
If  $t<0,$  then    $2t^2+t+a=0$    …(iii)
If Eq. (i) has four roots then Eq. (ii) must have both roots positive and Eq. (iii) has both roots negative. Now, Eq. (ii) has both roots positive, if D > 0 
  $a/2>0$
   $\Rightarrow 1-8a>0,a>0$
$\Rightarrow a\in (0,\frac{1}{8})$   on taking intersection.
Again, Eq. (iii) has both roots negative, if  $D>0,a/2>0.$
We again get  $a\in (0,\frac{1}{8})\Rightarrow K=8$