Solve x^2+x^4+5=0

$$x ^ { 2 } + x ^ { 4 } + 5 = 0$$

Solve for x (complex solution)

$$x=\sqrt[4]{5}e^{\frac{\left(\arctan(\sqrt{19})+\pi \right)i}{2}}\approx -0.931683417+1.169629851i$$

$$x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i+3\pi i}{2}}\approx 0.931683417-1.169629851i$$

$$x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+3\pi i}{2}}\approx -0.931683417-1.169629851i$$

$$x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+\pi i}{2}}\approx 0.931683417+1.169629851i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}+t+5=0$$

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 5 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\times 5}}{2}$$

Do the calculations.

$$t=\frac{-1±\sqrt{-19}}{2}$$

Solve the equation t=\frac{-1±\sqrt{-19}}{2} when ± is plus and when ± is minus.

$$t=\frac{-1+\sqrt{19}i}{2} t=\frac{-\sqrt{19}i-1}{2}$$

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

$$x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+3\pi i}{2}} x=\sqrt[4]{5}e^{\frac{-\arctan(\sqrt{19})i+\pi i}{2}} x=\sqrt[4]{5}e^{\frac{\arctan(\sqrt{19})i+3\pi i}{2}} x=\sqrt[4]{5}e^{\frac{\left(\arctan(\sqrt{19})+\pi \right)i}{2}}$$