x^3+x^2+54=34x find x

asked by guest
on Jan 21, 2025 at 8:16 pm



You asked:

Solve the equation \[{x}^{3} + {x}^{2} + 54 = 34 x\] for the unknown \(x\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{3} - \frac{2 \sqrt{103} \cos{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{3} \approx -6.9797149\\x &= - \frac{1}{3} + \frac{\sqrt{309} \sin{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} + \frac{\sqrt{103} \cos{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} - \frac{103 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{883 + 3 \sqrt{34782} i}}\right)}}{3} + i \left(- \frac{\sqrt{309} \cos{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} + \frac{\sqrt{103} \sin{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} - \frac{103 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{883 + 3 \sqrt{34782} i}}\right)}}{3}\right) \approx 4.0864622 + 2.0 \cdot 10^{-142} i\\x &= - \frac{\sqrt{309} \sin{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} - \frac{1}{3} - \frac{103 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{883 + 3 \sqrt{34782} i}}\right)}}{3} + \frac{\sqrt{103} \cos{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} + i \left(- \frac{103 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{883 + 3 \sqrt{34782} i}}\right)}}{3} + \frac{\sqrt{103} \sin{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6} + \frac{\sqrt{309} \cos{\left(\frac{\arctan{\left(\frac{3 \sqrt{34782}}{883} \right)}}{3} \right)}}{6}\right) \approx 1.8932527 + 4.0 \cdot 10^{-143} i\end{aligned}\]

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

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