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

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

Solve for x (complex solution)

$$x=\sqrt{2}e^{\frac{5\pi i}{6}}\approx -1.224744871+0.707106781i$$

$$x=\sqrt{2}e^{\frac{11\pi i}{6}}\approx 1.224744871-0.707106781i$$

$$x=\sqrt{2}e^{\frac{7\pi i}{6}}\approx -1.224744871-0.707106781i$$

$$x=\sqrt{2}e^{\frac{\pi i}{6}}\approx 1.224744871+0.707106781i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-2t+4=0$$

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, -2 for b, and 4 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{2±\sqrt{-12}}{2}$$

Solve the equation t=\frac{2±\sqrt{-12}}{2} when ± is plus and when ± is minus.

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

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

$$x=-\sqrt{2}e^{\frac{\pi i}{6}} x=\sqrt{2}e^{\frac{\pi i}{6}} x=-\sqrt{2}ie^{\frac{\pi i}{3}} x=\sqrt{2}ie^{\frac{\pi i}{3}}$$