Solve {l}{d=2}{fg=f_{0}(sqrt{1+(d/2)^2}pmd/2)}

$$\left. \begin{array} { l } { d = 2 } \\ { f g = f _ { 0 } ( \sqrt { 1 + ( \frac { d } { 2 } ) ^ { 2 } } \pm \frac { d } { 2 } ) } \end{array} \right.$$

Solve for d, f, g, f_0

$$\left\{\begin{matrix}\\d=2\text{, }f=\frac{f_{0}\left(\sqrt{2}±1\right)}{g}\text{, }g\neq 0\text{, }f_{0}\in \mathrm{R}\text{; }d=2\text{, }f\in \mathrm{R}\text{, }g=0\text{, }f_{0}=0\text{, }&\text{unconditionally}\\d=2\text{, }f\in \mathrm{R}\text{, }g=0\text{, }f_{0}\in \mathrm{R}\text{, }&\sqrt{2}±1=0\end{matrix}\right.$$