Solve x^4-4x^2+8=0

$${ x }^{ 4 } -4 { x }^{ 2 } +8=0$$

Solve for x (complex solution)

$$x=2^{\frac{3}{4}}e^{\frac{7\pi i}{8}}\approx -1.553773974+0.643594253i$$

$$x=2^{\frac{3}{4}}e^{\frac{15\pi i}{8}}\approx 1.553773974-0.643594253i$$

$$x=2^{\frac{3}{4}}e^{\frac{9\pi i}{8}}\approx -1.553773974-0.643594253i$$

$$x=2^{\frac{3}{4}}e^{\frac{\pi i}{8}}\approx 1.553773974+0.643594253i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-4t+8=0$$

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, -4 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$t=\frac{4±\sqrt{-16}}{2}$$

Solve the equation t=\frac{4±\sqrt{-16}}{2} when ± is plus and when ± is minus.

$$t=2+2i t=2-2i$$

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

$$x=-2^{\frac{3}{4}}e^{\frac{\pi i}{8}} x=2^{\frac{3}{4}}e^{\frac{\pi i}{8}} x=-2^{\frac{3}{4}}ie^{\frac{3\pi i}{8}} x=2^{\frac{3}{4}}ie^{\frac{3\pi i}{8}}$$