x³-3x+1=0

x³-3x+1=0

asked by guest
on Nov 17, 2024 at 3:46 am



You asked:

Solve the system of equations:\[\begin{aligned}{x}^{3} - 3 x + 1 &= 0\\{x}^{3} - 3 x + 1 &= 0\end{aligned}\]

MathBot Answer:

The system of equations has \(3\) solutions.

\[x = - 2 \cos{\left(\frac{\pi}{9} \right)}\]\[x = - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\]\[x = \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\]

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

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