Solve sqrt{1+k^2}*|frac{frac{2sqrt{3}}{3}-2sqrt{3}k^2}{k^2+1/4}|=2

$$\sqrt { 1 + k ^ { 2 } } \times | \frac { \frac { 2 \sqrt { 3 } } { 3 } - 2 \sqrt { 3 } k ^ { 2 } } { k ^ { 2 } + \frac { 1 } { 4 } } | = 2$$

Solve for k

$$k=-\frac{\sqrt[4]{312}\sqrt{\sqrt{3}\sin(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})+\cos(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})}}{6}\approx -0.88240543$$

$$k=\frac{\sqrt[4]{312}\sqrt{\sqrt{3}\sin(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})+\cos(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})}}{6}\approx 0.88240543$$

$$k=\frac{\sqrt[4]{312}\sqrt{-\sqrt{3}\sin(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})+\cos(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})}}{6}\approx 0.357626518$$

$$k=-\frac{\sqrt[4]{312}\sqrt{-\sqrt{3}\sin(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})+\cos(\frac{\arccos(\frac{9\sqrt{78}}{208})}{3})}}{6}\approx -0.357626518$$