3x^3-9x^2+6x+2=0
You asked:
Solve the equation \(3 {x}^{3} - 9 {x}^{2} + 6 x + 2 = 0\) for the variable \(x\).
MathBot Answer:
The 3 solutions to the equation are: \[\begin{aligned}x &= 1 - \frac{1}{\sqrt[3]{9 + 3 \sqrt{6}}} - \frac{\sqrt[3]{9 + 3 \sqrt{6}}}{3} \approx -0.24001181\\x &= \frac{1}{2 \sqrt[3]{3 \sqrt{6} + 9}} + \frac{\sqrt[3]{3 \sqrt{6} + 9}}{6} + 1 + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{3 \sqrt{6} + 9}} + \frac{\sqrt{3} \sqrt[3]{3 \sqrt{6} + 9}}{6}\right) \approx 1.6200059 + 0.39143578 i\\x &= \frac{1}{2 \sqrt[3]{3 \sqrt{6} + 9}} + \frac{\sqrt[3]{3 \sqrt{6} + 9}}{6} + 1 + i \left(- \frac{\sqrt{3} \sqrt[3]{3 \sqrt{6} + 9}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{3 \sqrt{6} + 9}}\right) \approx 1.6200059 -0.39143578 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).