Exactly 3 solutions are demanded

This can be written as

$\left| (x-2)(x-3) \right| =\lambda x-7\lambda$

There will be three solution if the line $\lambda x-7\lambda$ touches the the parabola.

So $-x^2+5x-6= \lambda x-7\lambda$ should have one root. So $D=0$

${ x }^{ 2 }-x(\lambda -5)+(6-7\lambda )=0$

${ (\lambda -5 })^{ 2 }-4(6-7\lambda )=0$

$\lambda =-9\pm 4\sqrt { 5 }$

Also the line should cut Y-axis near $1/4$ . So slope ( $\lambda$ ) should be closer to $-1/28$ . So the required answer is $\lambda =-9 +4\sqrt { 5 }$ .