x³+x=1 x=?

asked by guest
on Nov 24, 2024 at 12:57 am



You asked:

Solve the equation \({x}^{3} + x = 1\) for the variable \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}} - \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} \approx 0.6823278\\x &= - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right) \approx -0.3411639 + 1.1615414 i\\x &= - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right) \approx -0.3411639 -1.1615414 i\end{aligned}\]

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

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