Square of Real Roots

If we apply the Quadratic Formula, the roots are:

$x = \frac{-4A \pm \sqrt{16A^2 - 4(1)(5-A^2)}}{2} = \frac{-4A \pm \sqrt{20A^2-20}}{2} = -2A \pm \sqrt{5A^2-5}$

which the roots $x=B$ and $x=C$ will be minimal if they are real and repeated. This occurs when the discriminant equals zero, or at $A^2=1 \Rightarrow B = C = \pm 2.$ The smallest possible value of $B^2+C^2$ equals $(\pm2)^2 + (\pm2)^2 = 4+4 = \boxed{8}.$