Solve sqrt{16x^2+4(k^2-2k+1)-3}=sqrt{6}

$$\sqrt { 16 x ^ { 2 } + 4 ( k ^ { 2 } - 2 k + 1 ) - 3 } = \sqrt { 6 }$$

Solve for k (complex solution)

$$k=\frac{\sqrt{9-16x^{2}}}{2}+1$$

$$k=-\frac{\sqrt{9-16x^{2}}}{2}+1$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{5+8k-4k^{2}}}{4}$$

$$x=\frac{\sqrt{5+8k-4k^{2}}}{4}$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{\sqrt{9-16x^{2}}}{2}+1\text{, }&\left(\frac{-\sqrt{3-16x^{2}}-\sqrt{9-16x^{2}}}{2}\geq 0\text{ and }|x|\leq \frac{\sqrt{3}}{4}\right)\text{ or }\left(\frac{\sqrt{3-16x^{2}}-\sqrt{9-16x^{2}}}{2}\leq 0\text{ and }|x|\leq \frac{\sqrt{3}}{4}\right)\text{ or }\left(x>-\frac{\sqrt{6}}{4}\text{ and }x\leq -\frac{\sqrt{3}}{4}\right)\text{ or }\left(x\geq -\frac{3}{4}\text{ and }x\leq -\frac{\sqrt{6}}{4}\right)\text{ or }\left(x\geq \frac{\sqrt{6}}{4}\text{ and }x\leq \frac{3}{4}\right)\text{ or }\left(x\geq \frac{\sqrt{3}}{4}\text{ and }x<\frac{\sqrt{6}}{4}\right)\\k=\frac{\sqrt{9-16x^{2}}}{2}+1\text{, }&\left(\frac{\sqrt{9-16x^{2}}-\sqrt{3-16x^{2}}}{2}\geq 0\text{ and }|x|\leq \frac{\sqrt{3}}{4}\right)\text{ or }\left(\frac{\sqrt{3-16x^{2}}+\sqrt{9-16x^{2}}}{2}\leq 0\text{ and }|x|\leq \frac{\sqrt{3}}{4}\right)\text{ or }\left(x>-\frac{\sqrt{6}}{4}\text{ and }x\leq -\frac{\sqrt{3}}{4}\right)\text{ or }\left(x\geq -\frac{3}{4}\text{ and }x\leq -\frac{\sqrt{6}}{4}\right)\text{ or }\left(x\geq \frac{\sqrt{6}}{4}\text{ and }x\leq \frac{3}{4}\right)\text{ or }\left(x\geq \frac{\sqrt{3}}{4}\text{ and }x<\frac{\sqrt{6}}{4}\right)\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{5+8k-4k^{2}}}{4}\text{, }&\left(\frac{\sqrt{-4k^{2}+8k-1}-\sqrt{5+8k-4k^{2}}}{4}\leq 0\text{ or }\frac{-\sqrt{5+8k-4k^{2}}-\sqrt{-4k^{2}+8k-1}}{4}\geq 0\text{ or }k>\frac{\sqrt{3}}{2}+1\text{ or }k<-\frac{\sqrt{3}}{2}+1\right)\text{ and }k\geq -\frac{1}{2}\text{ and }k\leq \frac{5}{2}\\x=\frac{\sqrt{5+8k-4k^{2}}}{4}\text{, }&\left(\frac{\sqrt{5+8k-4k^{2}}+\sqrt{-4k^{2}+8k-1}}{4}\leq 0\text{ or }\frac{\sqrt{5+8k-4k^{2}}-\sqrt{-4k^{2}+8k-1}}{4}\geq 0\text{ or }k>\frac{\sqrt{3}}{2}+1\text{ or }k<-\frac{\sqrt{3}}{2}+1\right)\text{ and }k\geq -\frac{1}{2}\text{ and }k\leq \frac{5}{2}\end{matrix}\right.$$