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.