Mathematics - Quadratic Equation Question with Solution | TestHub

On putting ${\log }_{3}x=t$, we get

$2t^2-|t|+a=0$ ...............(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) both roots negative. Now, Eq. (ii) has both roots positive if $D>0$.

a /2 > 0

$\Rightarrow 1-8a>0\Rightarrow a<\frac{1}{8}$

On taking the intersection

Again, Eq. (iii) has both roots negative if $D>0$ and $a/2>0$.

We again get $a\in \left(0,\frac{1}{8}\right)\Rightarrow K=8$.