Solve y=frac{2kfrac{2k^{2}-4}{1+2k^{2}}+frac{4b^{2}}{1+2k^{2}}*k-4k}{2k^2-4/1+2k^2+24k^2/1+2k^2+4}

$$y = \frac { 2 k \frac { 2 k ^ { 2 } - 4 } { 1 + 2 k ^ { 2 } } + \frac { 4 b ^ { 2 } } { 1 + 2 k ^ { 2 } } \cdot k - 4 k } { \frac { 2 k ^ { 2 } - 4 } { 1 + 2 k ^ { 2 } } + 2 \frac { 4 k ^ { 2 } } { 1 + 2 k ^ { 2 } } + 4 }$$

Solve for b (complex solution)

$$b=-\frac{\sqrt{18ky+4k^{2}+12}}{2}$$

$$b=\frac{\sqrt{18ky+4k^{2}+12}}{2}\text{, }k\neq -\frac{\sqrt{2}i}{2}\text{ and }k\neq \frac{\sqrt{2}i}{2}\text{ and }k\neq 0$$

Solve for b

$$b=\frac{\sqrt{18ky+4k^{2}+12}}{2}$$

$$b=-\frac{\sqrt{18ky+4k^{2}+12}}{2}\text{, }\left(k>0\text{ or }y\leq -\frac{2k}{9}-\frac{2}{3k}\right)\text{ and }\left(k<0\text{ or }y\geq -\frac{2k}{9}-\frac{2}{3k}\right)\text{ and }k\neq 0$$

Solve for k

$$\left\{\begin{matrix}k=\frac{-\sqrt{81y^{2}+16b^{2}-48}-9y}{4}\text{, }&\left(y>0\text{ and }y\geq \frac{4\sqrt{3-b^{2}}}{9}\text{ and }|b|\leq \sqrt{3}\right)\text{ or }\left(y\geq \frac{4\sqrt{3-b^{2}}}{9}\text{ and }y>0\text{ and }|b|\neq \sqrt{3}\right)\text{ or }\left(b\neq -\sqrt{3}\text{ and }|b|<\sqrt{3}\text{ and }|y|\geq \frac{4\sqrt{3-b^{2}}}{9}\right)\text{ or }|b|>\sqrt{3}\\k=\frac{\sqrt{81y^{2}+16b^{2}-48}-9y}{4}\text{, }&\left(y<0\text{ and }y\leq -\frac{4\sqrt{3-b^{2}}}{9}\text{ and }|b|\leq \sqrt{3}\right)\text{ or }\left(y\leq -\frac{4\sqrt{3-b^{2}}}{9}\text{ and }y<0\text{ and }|b|\neq \sqrt{3}\right)\text{ or }\left(b\neq -\sqrt{3}\text{ and }|b|<\sqrt{3}\text{ and }|y|\geq \frac{4\sqrt{3-b^{2}}}{9}\right)\text{ or }|b|>\sqrt{3}\end{matrix}\right.$$