4x^3-14x-8=0 का समाधान करें

$$4 x ^ { 3 } - 14 x - 8 = 0$$

x के लिए हल करें (जटिल समाधान)

$$x=-\frac{36^{\frac{2}{3}}\times 972^{\frac{5}{6}}\left(1+\sqrt{3}i\right)\left(36+\sqrt{762}i\right)^{-\frac{1}{3}}\left(3\left(36+\sqrt{762}i\right)^{\frac{2}{3}}\left(-36^{\frac{2}{3}}i-2\sqrt[6]{972}\right)+14\times 36^{\frac{2}{3}}\sqrt[6]{972}\right)}{839808}\approx -1.4592613+1.386213537 \cdot 10^{-16}i$$

$$x=\frac{36^{\frac{2}{3}}\left(36+\sqrt{762}i\right)^{-\frac{1}{3}}\left(6\left(36+\sqrt{762}i\right)^{\frac{2}{3}}+7\times 36^{\frac{2}{3}}\right)}{216}\approx 2.109093351$$

$$x=-\frac{36^{\frac{2}{3}}\times 972^{\frac{5}{6}}\left(36+\sqrt{762}i\right)^{-\frac{1}{3}}\left(-\sqrt{3}i+1\right)\left(3\left(36+\sqrt{762}i\right)^{\frac{2}{3}}\left(36^{\frac{2}{3}}i-2\sqrt[6]{972}\right)+14\times 36^{\frac{2}{3}}\sqrt[6]{972}\right)}{839808}\approx -0.649832052-6.931067686 \cdot 10^{-17}i$$

x के लिए हल करें

$$x=-\frac{\sqrt{42}\left(\sqrt{3}\sin(\frac{\arccos(\frac{6\sqrt{42}}{49})}{3})+\cos(\frac{\arccos(\frac{6\sqrt{42}}{49})}{3})\right)}{6}\approx -1.4592613$$

$$x=\frac{\sqrt{42}\cos(\frac{\arccos(\frac{6\sqrt{42}}{49})}{3})}{3}\approx 2.109093351$$

$$x=\frac{\sqrt{42}\left(\sqrt{3}\sin(\frac{\arccos(\frac{6\sqrt{42}}{49})}{3})-\cos(\frac{\arccos(\frac{6\sqrt{42}}{49})}{3})\right)}{6}\approx -0.649832052$$