Solve x^4+x^2+27=0

$$x ^ { 4 } + x ^ { 2 } + 27 = 0$$

Solve for x (complex solution)

$$x=3^{\frac{3}{4}}e^{\frac{\left(\arctan(\sqrt{107})+\pi \right)i}{2}}\approx -1.532343373+1.687624428i$$

$$x=3^{\frac{3}{4}}e^{\frac{\arctan(\sqrt{107})i+3\pi i}{2}}\approx 1.532343373-1.687624428i$$

$$x=3^{\frac{3}{4}}e^{\frac{-\arctan(\sqrt{107})i+3\pi i}{2}}\approx -1.532343373-1.687624428i$$

$$x=3^{\frac{3}{4}}e^{\frac{-\arctan(\sqrt{107})i+\pi i}{2}}\approx 1.532343373+1.687624428i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}+t+27=0$$

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 27 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 1\times 27}}{2}$$

Do the calculations.

$$t=\frac{-1±\sqrt{-107}}{2}$$

Solve the equation t=\frac{-1±\sqrt{-107}}{2} when ± is plus and when ± is minus.

$$t=\frac{-1+\sqrt{107}i}{2} t=\frac{-\sqrt{107}i-1}{2}$$

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

$$x=3^{\frac{3}{4}}e^{\frac{-\arctan(\sqrt{107})i+3\pi i}{2}} x=3^{\frac{3}{4}}e^{\frac{-\arctan(\sqrt{107})i+\pi i}{2}} x=3^{\frac{3}{4}}e^{\frac{\arctan(\sqrt{107})i+3\pi i}{2}} x=3^{\frac{3}{4}}e^{\frac{\left(\arctan(\sqrt{107})+\pi \right)i}{2}}$$