x^4 +1 =0

asked by guest
on Nov 23, 2024 at 10:35 am



You asked:

Solve the equation \({x}^{4} + 1 = 0\) for the variable \(x\).

MathBot Answer:

The solutions are given by the formula: \[x = \cos{\left(\frac{\pi}{4} + \frac{\pi k}{2} \right)} + \sin{\left(\frac{\pi}{4} + \frac{\pi k}{2} \right)} i\] where \(k\) is an integer in \(0 \le k < 4\).

There are \(0\) real solutions.

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

The are no real solutions.

Solutions with nonzero imaginary part: \[\begin{aligned}x &= \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \\&\approx 0.70710678 + 0.70710678 i\\x &= - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} \\&\approx -0.70710678 + 0.70710678 i\\x &= - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \\&\approx -0.70710678 -0.70710678 i\\x &= \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} \\&\approx 0.70710678 -0.70710678 i\end{aligned}\]

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

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