x^3 - 5 + 3i = 0

asked by guest
on Jan 23, 2025 at 3:43 pm



You asked:

Investigate the equation: \({x}^{3} - 5 + 3 i = 0\).

MathBot Answer:

The 3 solutions to the equation are: \[\begin{aligned}x &= \sqrt[6]{34} \cos{\left(\frac{\arctan{\left(\frac{3}{5} \right)}}{3} \right)} - \sqrt[6]{34} i \sin{\left(\frac{\arctan{\left(\frac{3}{5} \right)}}{3} \right)} \approx 1.7707675 -0.32248154 i\\x &= - \sqrt[6]{34} \sin{\left(- \frac{\arctan{\left(\frac{3}{5} \right)}}{3} + \frac{\pi}{6} \right)} + \sqrt[6]{34} i \cos{\left(- \frac{\arctan{\left(\frac{3}{5} \right)}}{3} + \frac{\pi}{6} \right)} \approx -0.60610653 + 1.6947704 i\\x &= - \sqrt[6]{34} \cos{\left(- \frac{\arctan{\left(\frac{3}{5} \right)}}{3} + \frac{\pi}{3} \right)} - \sqrt[6]{34} i \sin{\left(- \frac{\arctan{\left(\frac{3}{5} \right)}}{3} + \frac{\pi}{3} \right)} \approx -1.1646609 -1.3722888 i\end{aligned}\]

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

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