Let $$S = \{\sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0$$ has real roots$$\}$$. If $$\alpha$$ and $$\beta$$ be the smallest and largest elements of the set $$S$$, respectively, then $$3((\alpha - 2)^2 + (\beta - 1)^2)$$ equals _____

Simplifying trigonometric terms with $$t = \sin^2 2\theta$$:

$$\left(1 - \frac{t}{2}\right)x^2 + (\sin 2\theta)x + \left(1 - \frac{3t}{4}\right) = 0$$

For real roots, Discriminant $$D \ge 0 \implies 3t^2 - 12t + 8 \le 0$$.

Solving gives $$t \in [2 - \frac{2}{\sqrt{3}}, 2 + \frac{2}{\sqrt{3}}]$$. Since $$0 \le t \le 1$$, the valid range is $$[2 - \frac{2}{\sqrt{3}}, 1]$$.

Thus, the smallest element $$\alpha = 2 - \frac{2}{\sqrt{3}}$$ and the largest element $$\beta = 1$$.

Substituting into $$3((\alpha - 2)^2 + (\beta - 1)^2)$$:

$$3\left(\left(-\frac{2}{\sqrt{3}}\right)^2 + 0\right) = 3\left(\frac{4}{3}\right) = 4$$