Solve w^4+w^2(7w_0^2+w_1^2)-2w_0^2w_1^2=0

$$w ^ { 4 } + w ^ { 2 } ( 7 w _ { 0 } ^ { 2 } + w _ { 1 } ^ { 2 } ) - 2 w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } = 0$$

Solve for w (complex solution)

$$w=\frac{\sqrt{-2\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-2w_{1}^{2}-14w_{0}^{2}}}{2}$$

$$w=-\frac{\sqrt{-2\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-2w_{1}^{2}-14w_{0}^{2}}}{2}$$

$$w=-\frac{\sqrt{2\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-2w_{1}^{2}-14w_{0}^{2}}}{2}$$

$$w=\frac{\sqrt{2\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-2w_{1}^{2}-14w_{0}^{2}}}{2}$$

Solve for w_0 (complex solution)

$$\left\{\begin{matrix}w_{0}=-i\left(7w^{2}-2w_{1}^{2}\right)^{-\frac{1}{2}}w\sqrt{w^{2}+w_{1}^{2}}\text{; }w_{0}=i\left(7w^{2}-2w_{1}^{2}\right)^{-\frac{1}{2}}w\sqrt{w^{2}+w_{1}^{2}}\text{, }&w\neq -\frac{\sqrt{14}w_{1}}{7}\text{ and }w\neq \frac{\sqrt{14}w_{1}}{7}\\w_{0}\in \mathrm{C}\text{, }&w=0\text{ and }w_{1}=0\end{matrix}\right.$$

Solve for w

$$\left\{\begin{matrix}\\w=-\frac{\sqrt{2\left(\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-7w_{0}^{2}-w_{1}^{2}\right)}}{2}\text{; }w=\frac{\sqrt{2\left(\sqrt{49w_{0}^{4}+w_{1}^{4}+22\left(w_{0}w_{1}\right)^{2}}-7w_{0}^{2}-w_{1}^{2}\right)}}{2}\text{, }&\text{unconditionally}\\w=0\text{, }&w_{1}=0\text{ and }w_{0}=0\end{matrix}\right.$$

Solve for w_0

$$\left\{\begin{matrix}w_{0}=\sqrt{-\frac{w^{2}\left(w^{2}+w_{1}^{2}\right)}{7w^{2}-2w_{1}^{2}}}\text{; }w_{0}=-\sqrt{-\frac{w^{2}\left(w^{2}+w_{1}^{2}\right)}{7w^{2}-2w_{1}^{2}}}\text{, }&|w|<\frac{\sqrt{14}|w_{1}|}{7}\\w_{0}\in \mathrm{R}\text{, }&w=0\text{ and }w_{1}=0\end{matrix}\right.$$