x^4+x^3+x^2+x+1=0
You asked:
Solve the equation \({x}^{4} + {x}^{3} + {x}^{2} + x + 1 = 0\) for the variable \(x\).
MathBot Answer:
The 4 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \approx 0.30901699 + 0.95105652 i\\x &= - \frac{\sqrt{5}}{4} - \frac{1}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \approx -0.80901699 + 0.58778525 i\\x &= - \frac{1}{4} + \frac{\sqrt{5}}{4} - i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \approx 0.30901699 -0.95105652 i\\x &= - \frac{\sqrt{5}}{4} - \frac{1}{4} - i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \approx -0.80901699 -0.58778525 i\end{aligned}\]
\(i\) is the imaginary unit, defined as \(i^2 = -1\).