Solve y=10k+12/sqrt{5k^2+10k+10}

$$y = \frac { 10 k + 12 } { \sqrt { 5 k ^ { 2 } + 10 k + 10 } }$$

Solve for k

$$\left\{\begin{matrix}k=\frac{-y\sqrt{5\left(104-5y^{2}\right)}-5y^{2}+120}{5\left(y^{2}-20\right)}\text{, }&y\leq \frac{2\sqrt{130}}{5}\text{ and }y\neq 2\sqrt{5}\text{ and }y>0\\k=\frac{\sqrt{5\left(104-5y^{2}\right)}y^{2}-5|y|y^{2}+120|y|}{5|y|\left(y^{2}-20\right)}\text{, }&\left(y>-2\sqrt{5}\text{ and }y<0\right)\text{ or }\left(y>2\sqrt{5}\text{ and }y\leq \frac{2\sqrt{130}}{5}\right)\\k=\frac{7}{5}=1.4\text{, }&y=2\sqrt{5}\\k\in \mathrm{R}\text{, }&|y|=2\sqrt{5}\text{ and }|y|=2\sqrt{6}\text{ and }|y|=\frac{6\sqrt{10}}{5}\\k=-\frac{6}{5}=-1.2\text{, }&\sqrt{5}y=0\end{matrix}\right.$$

Solve for y

$$y=\frac{2\sqrt{\frac{5}{k^{2}+2k+2}}\left(5k+6\right)}{5}$$