Solve (2+9k^2)x^2-54k^2x+(81k^2-18)=0

$$( 2 + 9 k ^ { 2 } ) x ^ { 2 } - 54 k ^ { 2 } x + ( 81 k ^ { 2 } - 18 ) = 0$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=-\frac{i\left(x-3\right)^{-\frac{1}{2}}\sqrt{2\left(x+3\right)}}{3}\text{; }k=\frac{i\left(x-3\right)^{-\frac{1}{2}}\sqrt{2\left(x+3\right)}}{3}\text{, }&x\neq 3\\k\in \mathrm{C}\text{, }&x=3\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=3\text{, }&\text{unconditionally}\\x=-\frac{3\left(2-9k^{2}\right)}{9k^{2}+2}\text{, }&k\neq -\frac{\sqrt{2}i}{3}\text{ and }k\neq \frac{\sqrt{2}i}{3}\end{matrix}\right.$$

Solve for k

$$\left\{\begin{matrix}k=\frac{\sqrt{-\frac{2\left(x+3\right)}{x-3}}}{3}\text{; }k=-\frac{\sqrt{-\frac{2\left(x+3\right)}{x-3}}}{3}\text{, }&x\geq -3\text{ and }x<3\\k\in \mathrm{R}\text{, }&x=3\end{matrix}\right.$$

Solve for x

$$x=-\frac{3\left(2-9k^{2}\right)}{9k^{2}+2}$$

$$x=3$$