2x=x(1+i)^5

asked by guest
on Nov 15, 2024 at 6:16 am



You asked:

Investigate the equation: \(2 x = {x\left( 1 + i \right)}^{5}\).

MathBot Answer:

The 5 solutions to the equation are: \[\begin{aligned}x &= 0\\x &= - \frac{2^{\frac{5}{8}} \sin{\left(\frac{3 \pi}{16} \right)}}{2} + \frac{2^{\frac{5}{8}} i \cos{\left(\frac{3 \pi}{16} \right)}}{2} \approx -0.42840321 + 0.64115072 i\\x &= \frac{2^{\frac{5}{8}} \sin{\left(\frac{3 \pi}{16} \right)}}{2} - \frac{2^{\frac{5}{8}} i \cos{\left(\frac{3 \pi}{16} \right)}}{2} \approx 0.42840321 -0.64115072 i\\x &= - \frac{2^{\frac{5}{8}} \cos{\left(\frac{3 \pi}{16} \right)}}{2} - \frac{2^{\frac{5}{8}} i \sin{\left(\frac{3 \pi}{16} \right)}}{2} \approx -0.64115072 -0.42840321 i\\x &= \frac{2^{\frac{5}{8}} \cos{\left(\frac{3 \pi}{16} \right)}}{2} + \frac{2^{\frac{5}{8}} i \sin{\left(\frac{3 \pi}{16} \right)}}{2} \approx 0.64115072 + 0.42840321 i\end{aligned}\]

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

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