Solve x^4+1/x^4=1

$${ x }^{ 4 } + \frac{ 1 }{ { x }^{ 4 } } = 1$$

Solve for x (complex solution)

$$x\in e^{\frac{13\pi i}{12}},e^{\frac{19\pi i}{12}},e^{\frac{7\pi i}{12}},e^{\frac{\pi i}{12}},e^{\frac{23\pi i}{12}},e^{\frac{17\pi i}{12}},e^{\frac{11\pi i}{12}},e^{\frac{5\pi i}{12}}$$

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^{4}.

$$x^{4}x^{4}+1=x^{4}$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$x^{8}+1=x^{4}$$

Subtract x^{4} from both sides.

$$x^{8}+1-x^{4}=0$$

Substitute t for x^{4}.

$$t^{2}-t+1=0$$

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 1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{1±\sqrt{-3}}{2}$$

Solve the equation t=\frac{1±\sqrt{-3}}{2} when ± is plus and when ± is minus.

$$t=\frac{1+\sqrt{3}i}{2} t=\frac{-\sqrt{3}i+1}{2}$$

Since x=t^{4}, the solutions are obtained by solving the equation for each t.

$$x=-ie^{\frac{\pi i}{12}} x=-e^{\frac{\pi i}{12}} x=ie^{\frac{\pi i}{12}} x=e^{\frac{\pi i}{12}} x=-ie^{\frac{5\pi i}{12}} x=-e^{\frac{5\pi i}{12}} x=ie^{\frac{5\pi i}{12}} x=e^{\frac{5\pi i}{12}}$$

Variable x cannot be equal to 0.

$$x=e^{\frac{5\pi i}{12}}\text{, }x\neq 0 x=ie^{\frac{5\pi i}{12}}\text{, }x\neq 0 x=-e^{\frac{5\pi i}{12}}\text{, }x\neq 0 x=-ie^{\frac{5\pi i}{12}}\text{, }x\neq 0 x=e^{\frac{\pi i}{12}}\text{, }x\neq 0 x=ie^{\frac{\pi i}{12}}\text{, }x\neq 0 x=-e^{\frac{\pi i}{12}}\text{, }x\neq 0 x=-ie^{\frac{\pi i}{12}}\text{, }x\neq 0$$