Solve left(1+k^2right)x^2+left(6k^2-2right)x+9k^2-3=0

$$\left( 1+ { k }^{ 2 } \right) { x }^{ 2 } + \left( 6 { k }^{ 2 } -2 \right) x+9 { k }^{ 2 } -3=0$$

Solve for k (complex solution)

$$k=-\frac{i\sqrt{x-3}\sqrt{x+1}}{x+3}$$

$$k=\frac{i\sqrt{x-3}\sqrt{x+1}}{x+3}\text{, }x\neq -3$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-3k^{2}+2\sqrt{1-3k^{2}}+1}{k^{2}+1}\text{; }x=\frac{-3k^{2}-2\sqrt{1-3k^{2}}+1}{k^{2}+1}\text{, }&k\neq -i\text{ and }k\neq i\\x=-\frac{3}{2}\text{, }&k=-i\text{ or }k=i\end{matrix}\right.$$

Solve for k

$$k=\frac{\sqrt{\left(3-x\right)\left(x+1\right)}}{x+3}$$

$$k=-\frac{\sqrt{\left(3-x\right)\left(x+1\right)}}{x+3}\text{, }x\geq -1\text{ and }x\leq 3$$

Solve for x

$$x=\frac{-3k^{2}+2\sqrt{1-3k^{2}}+1}{k^{2}+1}$$

$$x=\frac{-3k^{2}-2\sqrt{1-3k^{2}}+1}{k^{2}+1}\text{, }|k|\leq \frac{\sqrt{3}}{3}$$