# Hot And Fresh

**Algebra**Level 5

\[ 3 \left [\log_3 ( \sqrt{2+x} + \sqrt{2-x} ) \right]^2 + 2 \log_{1/3} ( \sqrt{2+x} + \sqrt{2-x} ) \cdot \log_3 (9x^2) + (1 - \log_{1/3} x )^2 = 0 \]

If the sum of the zero-free root(s) of the equation above can be expressed as \(\dfrac{a\sqrt{b}}{c}\) where \(a\), \(b\), and \(c\) are pairwise coprime positive integers, determine \(a+b+c\).

*This problem is extracted from the 2016 Vietnamese University Entrance Examination which just took place a few hours ago. Solutions and discussions are always welcome!*

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