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Algebra Level 4

A sequence of functions \(\{f_n(x) \}\) is defined recursively as follows:

\(\large \begin{align*} f_1(x) &= \sqrt {x^2 + 48}, \quad \text{and} \\ f_{n + 1}(x) &= \sqrt {x^2 + 6f_n(x)} \quad \text{for } n \geq 1. \end{align*}\).

For each positive integer \(n\), if there is only one real solution of the equation \(f_n(x) = 2x\), find that solution.

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