Solve n^2/2frac{sqrt{-8n^2-18(n^2-1)+16}}{(n^2-1)+2}=2/3

$$\frac { n ^ { 2 } } { 2 } \frac { \sqrt { - 8 n ^ { 2 } - 18 ( n ^ { 2 } - 1 ) + 16 } } { ( n ^ { 2 } - 1 ) + 2 } = \frac { 2 } { 3 }$$

Solve for n

$$n=-\frac{\sqrt{39\left(\sqrt{46227}\sin(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})-\sqrt{15409}\cos(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})+145\right)}}{117}\approx -0.608198338$$

$$n=\frac{\sqrt{39\left(\sqrt{46227}\sin(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})-\sqrt{15409}\cos(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})+145\right)}}{117}\approx 0.608198338$$

$$n=\frac{\sqrt{39\left(2\sqrt{15409}\cos(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})+145\right)}}{117}\approx 1.022795577$$

$$n=-\frac{\sqrt{39\left(2\sqrt{15409}\cos(\frac{\arccos(\frac{348733\sqrt{15409}}{237437281})}{3})+145\right)}}{117}\approx -1.022795577$$