Լուծեք x^3-4x+1=0

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

Լուծել x-ի համար (complex solution)

$$x=-\frac{18^{\frac{2}{3}}\times 3888^{\frac{5}{6}}\left(-9+\sqrt{687}i\right)^{-\frac{1}{3}}\left(1+\sqrt{3}i\right)\left(3\left(-9+\sqrt{687}i\right)^{\frac{2}{3}}\left(-18^{\frac{2}{3}}i-\sqrt[6]{3888}\right)+8\times 18^{\frac{2}{3}}\sqrt[6]{3888}\right)}{839808}\approx -2.114907541-1.386213537 \cdot 10^{-16}i$$

$$x=\frac{18^{\frac{2}{3}}\left(-9+\sqrt{687}i\right)^{-\frac{1}{3}}\left(3\left(-9+\sqrt{687}i\right)^{\frac{2}{3}}+4\times 18^{\frac{2}{3}}\right)}{54}\approx 1.860805853-1.315819881 \cdot 10^{-16}i$$

$$x=-\frac{18^{\frac{2}{3}}\times 3888^{\frac{5}{6}}\left(-9+\sqrt{687}i\right)^{-\frac{1}{3}}\left(-\sqrt{3}i+1\right)\left(3\left(-9+\sqrt{687}i\right)^{\frac{2}{3}}\left(18^{\frac{2}{3}}i-\sqrt[6]{3888}\right)+8\times 18^{\frac{2}{3}}\sqrt[6]{3888}\right)}{839808}\approx 0.254101688+2.295916171 \cdot 10^{-16}i$$

Լուծել x-ի համար

$$x=-\frac{4\sqrt{3}\cos(\frac{\arccos(\frac{3\sqrt{3}}{16})}{3})}{3}\approx -2.114907541$$

$$x=\frac{2\sqrt{3}\left(\sqrt{3}\sin(\frac{\arccos(\frac{3\sqrt{3}}{16})}{3})+\cos(\frac{\arccos(\frac{3\sqrt{3}}{16})}{3})\right)}{3}\approx 1.860805853$$

$$x=\frac{2\left(\sqrt{3}\cos(\frac{\arccos(\frac{3\sqrt{3}}{16})}{3})-3\sin(\frac{\arccos(\frac{3\sqrt{3}}{16})}{3})\right)}{3}\approx 0.254101688$$