다음을 풀어보세요: 27x+32x^2+7x^3+4=0

$$27 x + 32 x ^ { 2 } + 7 x ^ { 3 } + 4 = 0$$

x에 대한 해 (complex solution)

$$x=\frac{\sqrt{3}i\sqrt[3]{-8198+21\sqrt{64029}i}}{42}-\frac{457\sqrt{3}i\left(-8198+21\sqrt{64029}i\right)^{-\frac{1}{3}}}{42}-\frac{\sqrt[3]{-8198+21\sqrt{64029}i}}{42}-\frac{457\left(-8198+21\sqrt{64029}i\right)^{-\frac{1}{3}}}{42}-\frac{32}{21}\approx -3.52247241-5.551115123 \cdot 10^{-17}i$$

$$x=\frac{\sqrt[3]{-8198+21\sqrt{64029}i}+457\left(-8198+21\sqrt{64029}i\right)^{-\frac{1}{3}}-32}{21}\approx -0.188541273-3.383536837 \cdot 10^{-16}i$$

$$x=-\frac{\sqrt{3}i\sqrt[3]{-8198+21\sqrt{64029}i}}{42}+\frac{457\sqrt{3}i\left(-8198+21\sqrt{64029}i\right)^{-\frac{1}{3}}}{42}-\frac{\sqrt[3]{-8198+21\sqrt{64029}i}}{42}-\frac{457\left(-8198+21\sqrt{64029}i\right)^{-\frac{1}{3}}}{42}-\frac{32}{21}\approx -0.860414889+3.885780586 \cdot 10^{-16}i$$

x에 대한 해

$$x=\frac{2\left(-\sqrt{457}\cos(\frac{\arccos(\frac{8198\sqrt{457}}{208849})}{3})-16\right)}{21}\approx -3.52247241$$

$$x=\frac{\sqrt{457}\left(\sqrt{3}\sin(\frac{\arccos(\frac{8198\sqrt{457}}{208849})}{3})+\cos(\frac{\arccos(\frac{8198\sqrt{457}}{208849})}{3})\right)-32}{21}\approx -0.188541273$$

$$x=\frac{\sqrt{457}\left(-\sqrt{3}\sin(\frac{\arccos(\frac{8198\sqrt{457}}{208849})}{3})+\cos(\frac{\arccos(\frac{8198\sqrt{457}}{208849})}{3})\right)-32}{21}\approx -0.860414889$$