Golden Functions

Using the property, $(a + b)^2 - (a - b) ^2 = 4ab$

$(f(x) + g(x))^2 = (f(x) - g(x))^2 + 4f(x)g(x)$

$\Rightarrow (55\sqrt 5)^2 = x^2 + 4x$

$\Rightarrow x = \dfrac {-4 + \sqrt{16 - (-4\times 15125)}}{2} = -2 + \sqrt {15129} = -2 + 123 = \boxed{121}$

Note: I have considered positive sign for the square root term as we are looking for positive value of $x.$

$\begin{aligned} f(x) - g(x) & = x & \small \color{#3D99F6} \text{Squaring both sides} \\ f(x)^2 - 2{\color{#3D99F6}f(x) g(x)} + g(x)^2 & = x^2 & \small \color{#3D99F6} \text{Given that }f(x)g(x) = x \\ f(x)^2 - 2{\color{#3D99F6}x} + g(x)^2 & = x^2 \\ \implies f(x)^2 + g(x)^2 & = x^2 + 2x \end{aligned}$

Now we have:

$\begin{aligned} f(x) + g(x) & = 55\sqrt 5 & \small \color{#3D99F6} \text{Squaring both sides} \\ {\color{#3D99F6}f(x)^2} + 2f(x) g(x) + \color{#3D99F6}g(x)^2 & = 15125 & \small \color{#3D99F6} \text{Recall }f(x)^2+g(x)^2 = x^2 + 2x \\ {\color{#3D99F6}x^2 + 2x} + 2x & = 15125 \\ \implies x^2 + 4x - 15125 & = 0 \\ (x-121)(x+125) & = 0 \\ \implies x & = \boxed{121} & \small \color{#3D99F6} \text{SInce }x > 0 \end{aligned}$