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x^4+1=0

x^4+1=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 4        
x  + 1 = 0
x4+1=0x^{4} + 1 = 0
Simplify
Detail solution
Given the equation
x4+1=0x^{4} + 1 = 0
Because equation degree is equal to = 4 and the free term = -1 < 0,
so the real solutions of the equation d'not exist

All other 4 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z4=−1z^{4} = -1
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r4e4ip=−1r^{4} e^{4 i p} = -1
where
r=1r = 1
- the magnitude of the complex number
Substitute r:
e4ip=−1e^{4 i p} = -1
Using Euler’s formula, we find roots for p
isin⁡(4p)+cos⁡(4p)=−1i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = -1
so
cos⁡(4p)=−1\cos{\left(4 p \right)} = -1
and
sin⁡(4p)=0\sin{\left(4 p \right)} = 0
then
p=πN2+π4p = \frac{\pi N}{2} + \frac{\pi}{4}
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
z1=−22−2i2z_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
z2=−22+2i2z_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
z3=22−2i2z_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
z4=22+2i2z_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−22−2i2x_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
x2=−22+2i2x_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
x3=22−2i2x_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}
x4=22+2i2x_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}
The graph
-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
    ___       ___       ___       ___     ___       ___     ___       ___
  \/ 2    I*\/ 2      \/ 2    I*\/ 2    \/ 2    I*\/ 2    \/ 2    I*\/ 2 
- ----- - ------- + - ----- + ------- + ----- - ------- + ----- + -------
    2        2          2        2        2        2        2        2
((22−2i2)+((−22−2i2)+(−22+2i2)))+(22+2i2)\left(\left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) + \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right)\right)\right) + \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) 
=
0
00 
product
/    ___       ___\ /    ___       ___\ /  ___       ___\ /  ___       ___\
|  \/ 2    I*\/ 2 | |  \/ 2    I*\/ 2 | |\/ 2    I*\/ 2 | |\/ 2    I*\/ 2 |
|- ----- - -------|*|- ----- + -------|*|----- - -------|*|----- + -------|
\    2        2   / \    2        2   / \  2        2   / \  2        2   /
(−22−2i2)(−22+2i2)(22−2i2)(22+2i2)\left(- \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(- \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2}\right) \left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2}\right) 
=
1
11 
1
Rapid solution [src] 
         ___       ___
       \/ 2    I*\/ 2 
x1 = - ----- - -------
         2        2
x1=−22−2i2x_{1} = - \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} 
         ___       ___
       \/ 2    I*\/ 2 
x2 = - ----- + -------
         2        2
x2=−22+2i2x_{2} = - \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} 
       ___       ___
     \/ 2    I*\/ 2 
x3 = ----- - -------
       2        2
x3=22−2i2x_{3} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2} i}{2} 
       ___       ___
     \/ 2    I*\/ 2 
x4 = ----- + -------
       2        2
x4=22+2i2x_{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2} i}{2} 
x4 = sqrt(2)/2 + sqrt(2)*i/2
Numerical answer [src] 
x1 = -0.707106781186548 - 0.707106781186548*i
x2 = 0.707106781186548 + 0.707106781186548*i
x3 = -0.707106781186548 + 0.707106781186548*i
x4 = 0.707106781186548 - 0.707106781186548*i
x4 = 0.707106781186548 - 0.707106781186548*i
The graph
x^4+1=0 equation
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