Red, Blue and Greenish

$ABCD$ is a rectangle, $AB = 2, AD=1$ . $E$ is chosen so that the radius of the blue circle is twice the radius of the red circle. Find the radius of the green circle. If the green circle radius is a root of $f(x) = ax^3+bx^2+cx+d$ , where $a>0$ , $\gcd (a,b,c,d)=1$ , submit $f(2)$ .