Solve x^4-x^2+14=0

$$x ^ { 4 } - x ^ { 2 } + 14 = 0$$

Solve for x (complex solution)

$$x=\sqrt[4]{14}e^{\frac{-\arctan(\sqrt{55})i+2\pi i}{2}}\approx -1.456306525+1.273117706i$$

$$x=\sqrt[4]{14}e^{-\frac{\arctan(\sqrt{55})i}{2}}\approx 1.456306525-1.273117706i$$

$$x=\sqrt[4]{14}e^{\frac{\arctan(\sqrt{55})i+2\pi i}{2}}\approx -1.456306525-1.273117706i$$

$$x=\sqrt[4]{14}e^{\frac{\arctan(\sqrt{55})i}{2}}\approx 1.456306525+1.273117706i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}-t+14=0$$

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 14 for c in the quadratic formula.

$$t=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 14}}{2}$$

Do the calculations.

$$t=\frac{1±\sqrt{-55}}{2}$$

Solve the equation t=\frac{1±\sqrt{-55}}{2} when ± is plus and when ± is minus.

$$t=\frac{1+\sqrt{55}i}{2} t=\frac{-\sqrt{55}i+1}{2}$$

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

$$x=\sqrt[4]{14}e^{\frac{\arctan(\sqrt{55})i+2\pi i}{2}} x=\sqrt[4]{14}e^{\frac{\arctan(\sqrt{55})i}{2}} x=\sqrt[4]{14}e^{-\frac{\arctan(\sqrt{55})i}{2}} x=\sqrt[4]{14}e^{\frac{-\arctan(\sqrt{55})i+2\pi i}{2}}$$