a^3-13a^2+46a-33=0 حل کریں

$$a ^ { 3 } - 13 a ^ { 2 } + 46 a - 33 = 0$$

a کے لئے حل کریں (complex solution)

$$a=-\frac{\sqrt[3]{2}\left(-97+9\sqrt{1355}i\right)^{-\frac{1}{3}}\left(1+\sqrt{3}i\right)\left(\sqrt[3]{2}\left(-97+9\sqrt{1355}i\right)^{\frac{2}{3}}\left(-\sqrt{3}i-1\right)+13\times 2^{\frac{2}{3}}\sqrt[3]{-97+9\sqrt{1355}i}\left(-1+\sqrt{3}i\right)+124\right)}{24}\approx 0.957317216-1.480297366 \cdot 10^{-16}i$$

$$a=\frac{2^{\frac{2}{3}}\left(-97+9\sqrt{1355}i\right)^{-\frac{1}{3}}\left(13\sqrt[3]{2\left(-97+9\sqrt{1355}i\right)}+\left(-97+9\sqrt{1355}i\right)^{\frac{2}{3}}+31\times 2^{\frac{2}{3}}\right)}{6}\approx 7.357462537$$

$$a=-\frac{\sqrt[3]{2}\left(-97+9\sqrt{1355}i\right)^{-\frac{1}{3}}\left(-\sqrt{3}i+1\right)\left(13\times 2^{\frac{2}{3}}\sqrt[3]{-97+9\sqrt{1355}i}\left(-\sqrt{3}i-1\right)+\sqrt[3]{2}\left(-97+9\sqrt{1355}i\right)^{\frac{2}{3}}\left(-1+\sqrt{3}i\right)+124\right)}{24}\approx 4.685220247+7.401486831 \cdot 10^{-16}i$$

a کے لئے حل کریں

$$a=\frac{-2\sqrt{31}\cos(\frac{\arccos(\frac{97\sqrt{31}}{1922})}{3})+13}{3}\approx 0.957317216$$

$$a=\frac{\sqrt{31}\left(\sqrt{3}\sin(\frac{\arccos(\frac{97\sqrt{31}}{1922})}{3})+\cos(\frac{\arccos(\frac{97\sqrt{31}}{1922})}{3})\right)+13}{3}\approx 7.357462537$$

$$a=\frac{\sqrt{31}\left(-\sqrt{3}\sin(\frac{\arccos(\frac{97\sqrt{31}}{1922})}{3})+\cos(\frac{\arccos(\frac{97\sqrt{31}}{1922})}{3})\right)+13}{3}\approx 4.685220247$$