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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 - contains the even number 4 in the numerator, then
the equation has two real roots.
Get the root 4-th degree of the equation sides:
We get:
(1x+0)44=1\sqrt[4]{\left(1 x + 0\right)^{4}} = 1
(1x+0)44=−1\sqrt[4]{\left(1 x + 0\right)^{4}} = -1
or
x=1x = 1
x=−1x = -1
We get the answer: x = 1
We get the answer: x = -1
or
x1=−1x_{1} = -1
x2=1x_{2} = 1

All other 2 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=πN2p = \frac{\pi N}{2}
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=−1z_{1} = -1
z2=1z_{2} = 1
z3=−iz_{3} = - i
z4=iz_{4} = i
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−1x_{1} = -1
x2=1x_{2} = 1
x3=−ix_{3} = - i
x4=ix_{4} = i
The graph
05-15-10-51015-2000020000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1
Simplify 
x2 = 1
x2=1x_{2} = 1
Simplify 
x3 = -I
x3=−ix_{3} = - i
Simplify 
x4 = I
x4=ix_{4} = i
Simplify
Sum and product of roots [src] 
sum
0 - 1 + 1 - I + I
(((−1+0)+1)−i)+i\left(\left(\left(-1 + 0\right) + 1\right) - i\right) + i 
=
0
00 
product
1*-1*1*-I*I
i−i1(−1)1i - i 1 \left(-1\right) 1 
=
-1
−1-1 
-1
Numerical answer [src] 
x1 = -1.0*i
x2 = -1.0
x3 = 1.0*i
x4 = 1.0
x4 = 1.0
The graph
x^4-1=0 equation
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