(1/3+k^{2})x^2+(2sqrt{3}k-2k^2)x^2+k^2-2sqrt{3}k+2=0 Denklemini Çözme

$$( \frac { 1 } { 3 } + k ^ { 2 } ) x ^ { 2 } + ( 2 \sqrt { 3 } k - 2 k ^ { 2 } ) x ^ { 2 } + k ^ { 2 } - 2 \sqrt { 3 } k + 2 = 0$$

k için çözün (complex solution)

$$k=\frac{\sqrt{3}\left(\sqrt{\left(10x^{2}-3\right)\left(x^{2}-1\right)}-3x^{2}+3\right)}{3\left(1-x^{2}\right)}$$

$$k=\frac{\sqrt{3}\left(-\sqrt{\left(10x^{2}-3\right)\left(x^{2}-1\right)}-3x^{2}+3\right)}{3\left(1-x^{2}\right)}\text{, }x\neq 1\text{ and }x\neq -1$$

x için çözün (complex solution)

$$x=-i\left(-3k^{2}+6\sqrt{3}k+1\right)^{-\frac{1}{2}}\sqrt{3\left(k^{2}-2\sqrt{3}k+2\right)}$$

$$x=i\left(-3k^{2}+6\sqrt{3}k+1\right)^{-\frac{1}{2}}\sqrt{3\left(k^{2}-2\sqrt{3}k+2\right)}\text{, }k\neq -\frac{\sqrt{30}}{3}+\sqrt{3}\text{ and }k\neq \frac{\sqrt{30}}{3}+\sqrt{3}$$

k için çözün

$$k=\frac{\sqrt{3}\left(\sqrt{\left(10x^{2}-3\right)\left(x^{2}-1\right)}-3x^{2}+3\right)}{3\left(1-x^{2}\right)}$$

$$k=\frac{\sqrt{3}\left(-\sqrt{\left(10x^{2}-3\right)\left(x^{2}-1\right)}-3x^{2}+3\right)}{3\left(1-x^{2}\right)}\text{, }|x|\leq \frac{\sqrt{30}}{10}\text{ or }|x|>1$$

x için çözün

$$x=\sqrt{\frac{3\left(-k^{2}+2\sqrt{3}k-2\right)}{-3k^{2}+6\sqrt{3}k+1}}$$

$$x=-\sqrt{\frac{3\left(-k^{2}+2\sqrt{3}k-2\right)}{-3k^{2}+6\sqrt{3}k+1}}\text{, }\left(k\geq \sqrt{3}-1\text{ and }k\leq \sqrt{3}+1\right)\text{ or }k>\frac{\sqrt{30}}{3}+\sqrt{3}\text{ or }k<-\frac{\sqrt{30}}{3}+\sqrt{3}$$