Double Simplicity

Algebra Level 5

Let $$x,y$$ be real numbers such that $$x^2 + xy + 2y^2 = 8$$.

The greatest value that $$x + y$$ can attain is a real number equal to $$\dfrac{a\sqrt{b}}{b}$$, where $$a,b$$ are positive integers such that $$b$$ does not divide $$a$$, and $$b$$ is square- free.

Evaluate $$a+b$$.

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