Solve x^4+3x^2+1=0

$$x ^ { 4 } + 3 x ^ { 2 } + 1 = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{5}i}{2}+\frac{1}{2}i\approx 1.618033989i$$

$$x=-\frac{\sqrt{5}i}{2}-\frac{1}{2}i\approx -1.618033989i$$

$$x=-\frac{\sqrt{5}i}{2}+\frac{1}{2}i\approx -0.618033989i$$

$$x=\frac{\sqrt{5}i}{2}-\frac{1}{2}i\approx 0.618033989i$$

Steps Using the Quadratic Formula

Substitute t for x^{2}.

$$t^{2}+3t+1=0$$

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, 3 for b, and 1 for c in the quadratic formula.

$$t=\frac{-3±\sqrt{3^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{-3±\sqrt{5}}{2}$$

Solve the equation t=\frac{-3±\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{5}-3}{2} t=\frac{-\sqrt{5}-3}{2}$$

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

$$x=-\frac{\sqrt{5}i}{2}+\frac{1}{2}i x=-\left(-\frac{\sqrt{5}i}{2}+\frac{1}{2}i\right) x=-\left(\frac{\sqrt{5}i}{2}+\frac{1}{2}i\right) x=\frac{\sqrt{5}i}{2}+\frac{1}{2}i$$