求解 x^3+6x^2+9x+3=0

$${ x }^{ 3 } +6 { x }^{ 2 } +9x+3 = 0$$

求解 x 的值 (复数求解)

$$x=\frac{\left(-1+\sqrt{3}i\right)^{-\frac{4}{3}}\left(8\left(-\left(-1+\sqrt{3}i\right)^{\frac{4}{3}}+\sqrt[3]{2}\right)+2^{\frac{2}{3}}\left(-1+\sqrt{3}i\right)^{\frac{8}{3}}\right)}{4}\approx -3.879385242-2.220446049 \cdot 10^{-16}i$$

$$x=\frac{2^{\frac{2}{3}}\left(-1+\sqrt{3}i\right)^{-\frac{1}{3}}\left(\sqrt[3]{-1+\sqrt{3}i}-\sqrt[3]{2}\right)^{2}}{2}\approx -0.467911114-2.775557562 \cdot 10^{-17}i$$

$$x=\frac{\sqrt[3]{2}\left(1+\sqrt{3}i\right)^{-\frac{2}{3}}\left(\left(\left(1+\sqrt{3}i\right)^{\frac{2}{3}}+2^{\frac{2}{3}}\right)^{2}-4\times \left(2\left(1+\sqrt{3}i\right)\right)^{\frac{2}{3}}\right)}{2}\approx -0.467911114+4.996003611 \cdot 10^{-16}i$$

求解 x 的值

$$x=-4\left(\cos(\frac{\pi }{18})\right)^{2}\approx -3.879385242$$

$$x=-4\left(\sin(\frac{\pi }{9})\right)^{2}\approx -0.467911114$$

$$x=2\left(\sin(\frac{\pi }{18})-1\right)\approx -1.652703645$$