Solve x^2+5/x^2=1

$${ x }^{ 2 } + \frac{ 5 }{ { x }^{ 2 } } =1$$

Solve for x (complex solution)

$$x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i}{2}}\approx 1.169629851+0.931683417i$$

$$x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i+2\pi i}{2}}\approx -1.169629851-0.931683417i$$

$$x=\sqrt[4]{5}e^{-\frac{\arctan(\sqrt{19})i}{2}}\approx 1.169629851-0.931683417i$$

$$x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+2\pi i}{2}}\approx -1.169629851+0.931683417i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.

$$x^{2}x^{2}+5=x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{4}+5=x^{2}$$

Subtract x^{2} from both sides.

$$x^{4}+5-x^{2}=0$$

Substitute t for x^{2}.

$$t^{2}-t+5=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -1 for b, and 5 for c in the quadratic formula.

$$t=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 5}}{2}$$

Do the calculations.

$$t=\frac{1±\sqrt{-19}}{2}$$

Solve the equation t=\frac{1±\sqrt{-19}}{2} when ± is plus and when ± is minus.

$$t=\frac{1+\sqrt{19}i}{2} t=\frac{-\sqrt{19}i+1}{2}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i+2\pi i}{2}} x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i}{2}} x=\sqrt[4]{5}e^{-\frac{\arctan(\sqrt{19})i}{2}} x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+2\pi i}{2}}$$

Variable x cannot be equal to 0.

$$x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+2\pi i}{2}}\text{, }x\neq 0 x=\sqrt[4]{5}e^{-\frac{\arctan(\sqrt{19})i}{2}}\text{, }x\neq 0 x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i}{2}}\text{, }x\neq 0 x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i+2\pi i}{2}}\text{, }x\neq 0$$