Solve (4/k^2+3)x^2+16/k^2+16/k^2-48=0

$$( \frac { 4 } { k ^ { 2 } } + 3 ) x ^ { 2 } + \frac { 16 } { k ^ { 2 } } + \frac { 16 } { k ^ { 2 } } - 48 = 0$$

Solve for k (complex solution)

$$k=-\frac{2i\left(x^{2}-16\right)^{-\frac{1}{2}}\sqrt{3\left(x^{2}+8\right)}}{3}$$

$$k=\frac{2i\left(x^{2}-16\right)^{-\frac{1}{2}}\sqrt{3\left(x^{2}+8\right)}}{3}\text{, }x\neq -2\sqrt{2}i\text{ and }x\neq 2\sqrt{2}i\text{ and }x\neq 4\text{ and }x\neq -4$$

Solve for x (complex solution)

$$x=-4i\left(3k^{2}+4\right)^{-\frac{1}{2}}\sqrt{2-3k^{2}}$$

$$x=4i\left(3k^{2}+4\right)^{-\frac{1}{2}}\sqrt{2-3k^{2}}\text{, }k\neq -\frac{2\sqrt{3}i}{3}\text{ and }k\neq \frac{2\sqrt{3}i}{3}\text{ and }k\neq 0$$

Solve for k

$$k=\frac{2\sqrt{-\frac{3\left(x^{2}+8\right)}{x^{2}-16}}}{3}$$

$$k=-\frac{2\sqrt{-\frac{3\left(x^{2}+8\right)}{x^{2}-16}}}{3}\text{, }|x|<4$$

Solve for x

$$x=4\sqrt{\frac{3k^{2}-2}{3k^{2}+4}}$$

$$x=-4\sqrt{\frac{3k^{2}-2}{3k^{2}+4}}\text{, }|k|\geq \frac{\sqrt{6}}{3}\text{ and }k\neq 0$$