Given that, $A=90°$.

Circumradius $\left(R\right)=\frac{a}{2\sin A}=\frac{a}{2\sin 90°}=\frac{a}{2}$.

Inradius $\left(r\right)=\left(s-a\right)\tan \frac{A}{2}=\left(s-a\right)\tan \frac{90°}{2}=s-a$.

Exradius $\left(r_1\right)=s\tan \frac{A}{2}=s\tan \frac{90°}{2}=s$

$R:r:r_1=\frac{a}{2}:s-a:s=2:5:\lambda$

$\Rightarrow \frac{a}{2}=2\Rightarrow a=4$

$\Rightarrow \left(s-a\right)=5\Rightarrow s-4=5\Rightarrow s=9$

$\therefore \lambda =9$

Now, the equation is $x^2-\left(9-5\right)x+\left(9-6\right)=0$.

$\Rightarrow x^2-4x+3=0$

$\Rightarrow (x-1)(x-3)=0$

$\Rightarrow x=1$ or $x=3$

$\therefore$ The roots are $1, 3$.