Solve 15*4*k^{2}*x^2=25x^2*4(4R^2-25x^2)

$$15 \cdot 4 \cdot k ^ { 2 } \cdot x ^ { 2 } = 25 x ^ { 2 } \cdot 4 ( 4 R ^ { 2 } - 25 x ^ { 2 } )$$

Solve for R (complex solution)

$$\left\{\begin{matrix}\\R=-\frac{\sqrt{625x^{2}+15k^{2}}}{10}\text{; }R=\frac{\sqrt{625x^{2}+15k^{2}}}{10}\text{, }&\text{unconditionally}\\R\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Solve for k (complex solution)

$$\left\{\begin{matrix}\\k=-\frac{\sqrt{60R^{2}-375x^{2}}}{3}\text{; }k=\frac{\sqrt{60R^{2}-375x^{2}}}{3}\text{, }&\text{unconditionally}\\k\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Solve for R

$$\left\{\begin{matrix}\\R=\frac{\sqrt{625x^{2}+15k^{2}}}{10}\text{; }R=-\frac{\sqrt{625x^{2}+15k^{2}}}{10}\text{, }&\text{unconditionally}\\R\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Solve for k

$$\left\{\begin{matrix}k=\frac{\sqrt{60R^{2}-375x^{2}}}{3}\text{; }k=-\frac{\sqrt{60R^{2}-375x^{2}}}{3}\text{, }&|x|\leq \frac{2|R|}{5}\\k\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$