Infinite nested radical evaluation

Let $y = \sqrt{4 - x}$. Then $x = \sqrt{4 + y}$. Squaring both equations: $x^2 = 4 + y$ and $y^2 = 4 - x$. Subtracting: $x^2 - y^2 = x + y \implies (x - y)(x + y) = x + y$. Since $x + y \ne 0$, $x - y = 1 \implies y = x - 1$. Substitute into $x^2 = 4 + y = 4 + x - 1 = 3 + x$. $x^2 - x - 3 = 0$. By quadratic formula: $x = \frac{1 + \sqrt{1 + 12}}{2} = \frac{\sqrt{13} + 1}{2}$.