2=(1+x)^5

asked by guest
on Nov 15, 2024 at 6:17 am



You asked:

Solve the equation \(2 = {\left( 1 + x \right)}^{5}\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \left(-1 + \sqrt[5]{2} \cos{\left(\frac{2 \pi k}{5} \right)}\right) + \sqrt[5]{2} \sin{\left(\frac{2 \pi k}{5} \right)} i\] where \(k\) is an integer in \(0 \le k < 5\).

There is \(1\) real solution.

There are \(4\) solutions with nonzero imaginary part.

Real solutions: \[\begin{aligned}x &= -1 + \sqrt[5]{2} \\&\approx 0.14869836\end{aligned}\]

Solutions with nonzero imaginary part: \[\begin{aligned}x &= -1 + \sqrt[5]{2} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) + \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx -0.64503269 + 1.0924771 i\\x &= -1 + \sqrt[5]{2} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) + \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -1.9293165 + 0.67518795 i\\x &= -1 + \sqrt[5]{2} \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) - \sqrt[5]{2} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \\&\approx -1.9293165 -0.67518795 i\\x &= -1 + \sqrt[5]{2} \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) - \sqrt[5]{2} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \\&\approx -0.64503269 -1.0924771 i\end{aligned}\]

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

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