x^3+3x+2=0
You asked:
Solve the equation \({x}^{3} + 3 x + 2 = 0\) for the variable \(x\).
MathBot Answer:
The 3 solutions to the equation are: \[\begin{aligned}x &= \frac{3}{\sqrt[3]{27 + 27 \sqrt{2}}} - \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{3} \approx -0.59607164\\x &= - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6}\right) \approx 0.29803582 + 1.8073395 i\\x &= - \frac{3}{2 \sqrt[3]{27 + 27 \sqrt{2}}} + \frac{\sqrt[3]{27 + 27 \sqrt{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + 27 \sqrt{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{27 + 27 \sqrt{2}}}\right) \approx 0.29803582 -1.8073395 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).