(x-2)^4 - 1 = 0
You asked:
Solve the equation \({\left( x - 2 \right)}^{4} - 1 = 0\) for the variable \(x\).
MathBot Answer:
The solutions are given by the formula: \[x = \left(2 + \cos{\left(\frac{\pi k}{2} \right)}\right) + \sin{\left(\frac{\pi k}{2} \right)} i\] where \(k\) is an integer in \(0 \le k < 4\).
There are \(2\) real solutions.
There are \(2\) solutions with nonzero imaginary part.
Real solutions: \[\begin{aligned}x &= 3\\x &= 1\end{aligned}\]
Solutions with nonzero imaginary part: \[\begin{aligned}x &= 2 + i\\x &= 2 - i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).