Solve for x:
x^3+3 x^2+3 x+3 = 0
Subtract 2 from both sides:
x^3+3 x^2+3 x+1 = -2
Factor x^3+3 x^2+3 x+1 into a perfect cube:
(x+1)^3 = -2
Taking cube roots gives (-2)^(1/3) times the third roots of unity:
x+1 = (-2)^(1/3) or x+1 = -2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
x = (-2)^(1/3)-1 or x+1 = -2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
x = (-2)^(1/3)-1 or x = -1-2^(1/3) or x+1 = -((-1)^(2/3) 2^(1/3))
Subtract 1 from both sides:
Answer: |x = (-2)^(1/3)-1 or x = -1-2^(1/3) or x = -1-(-1)^(2/3) 2^(1/3)
+26388 roots of equation x^3+3x^2+3x+3
\(\small{ \begin{array}{|lrcll|} \hline & x^3+3x^2+3x+3 &=& 0 \quad &|\quad x^3+3x^2+3x+1 = (x+1)^3 \\ & (x+1)^3 +2 &=& 0 \\ & (x+1)^3 &=& -2 \\ & x+1 &=& \sqrt[3]{-2} \\ & x &=& \sqrt[3]{-2} -1 \\\\ (1) & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\cos(\frac{\pi}{3}) + i \cdot \sin(\frac{\pi}{3})) \quad &|\quad \tan{\varphi} = \frac{0}{-2}~ \Rightarrow ~ \varphi = \pi \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\frac12+ i \cdot \frac{\sqrt{3}}{2} ) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot \frac12+ i \cdot \frac{\sqrt{3}}{2} \cdot \sqrt[3]{2} \\ & x_1 = \sqrt[3]{-2}-1 &=& \sqrt[3]{2} \cdot \frac12 - 1 + i \cdot \frac{\sqrt{3}}{2} \cdot \sqrt[3]{2} \\ & \mathbf{x_1} &\mathbf{=}& \mathbf{-0.37003947505 + i \cdot 1.09112363597 }\\\\ (2) & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\cos(\frac{\pi}{3}+\frac23 \pi) + i \cdot \sin(\frac{\pi}{3}+\frac23 \pi)) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\cos(\pi) + i \cdot \sin(\pi)) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (-1+ i \cdot 0 ) \\ & \sqrt[3]{-2} &=& -\sqrt[3]{2} \\ & x_2 = \sqrt[3]{-2}-1 &=&-\sqrt[3]{2} - 1 \\ & \mathbf{x_2} &\mathbf{=}& \mathbf{-2.25992104989} \\\\ (3) & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\cos(\frac{\pi}{3}+2\cdot \frac23 \pi) + i \cdot \sin(\frac{\pi}{3}+2\cdot \frac23 \pi)) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\cos(\frac53 \pi) + i \cdot \sin(\frac53 \pi)) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot (\frac12- i \cdot \frac{\sqrt{3}}{2} ) \\ & \sqrt[3]{-2} &=& \sqrt[3]{2} \cdot \frac12- i \cdot \frac{\sqrt{3}}{2} \cdot \sqrt[3]{2} \\ & x_3 = \sqrt[3]{-2}-1 &=& \sqrt[3]{2} \cdot \frac12 - 1 - i \cdot \frac{\sqrt{3}}{2} \cdot \sqrt[3]{2} \\ & \mathbf{x_3} &\mathbf{=}& \mathbf{-0.37003947505 - i \cdot 1.09112363597} \\\\ \hline \end{array} } \)
