Solve x^4-x^2+1=0

$${ x }^{ 4 } - { x }^{ 2 } +1=0$$

Solve for x (complex solution)

$$x=e^{\frac{5\pi i}{6}}\approx -0.866025404+0.5i$$

$$x=e^{\frac{11\pi i}{6}}\approx 0.866025404-0.5i$$

$$x=e^{\frac{7\pi i}{6}}\approx -0.866025404-0.5i$$

$$x=e^{\frac{\pi i}{6}}\approx 0.866025404+0.5i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

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

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

$$x=-e^{\frac{\pi i}{6}} x=e^{\frac{\pi i}{6}} x=-ie^{\frac{\pi i}{3}} x=ie^{\frac{\pi i}{3}}$$