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x^4+4=0

x^4+4=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 4        
x  + 4 = 0
x4+4=0x^{4} + 4 = 0
Simplify
Detail solution
Given the equation
x4+4=0x^{4} + 4 = 0
Because equation degree is equal to = 4 and the free term = -4 < 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=−4z^{4} = -4
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r4e4ip=−4r^{4} e^{4 i p} = -4
where
r=2r = \sqrt{2}
- 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=−1−iz_{1} = -1 - i
z2=−1+iz_{2} = -1 + i
z3=1−iz_{3} = 1 - i
z4=1+iz_{4} = 1 + i
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−1−ix_{1} = -1 - i
x2=−1+ix_{2} = -1 + i
x3=1−ix_{3} = 1 - i
x4=1+ix_{4} = 1 + i
The graph
-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
-1 - I + -1 + I + 1 - I + 1 + I
((1−i)+((−1−i)+(−1+i)))+(1+i)\left(\left(1 - i\right) + \left(\left(-1 - i\right) + \left(-1 + i\right)\right)\right) + \left(1 + i\right) 
=
0
00 
product
(-1 - I)*(-1 + I)*(1 - I)*(1 + I)
(−1−i)(−1+i)(1−i)(1+i)\left(-1 - i\right) \left(-1 + i\right) \left(1 - i\right) \left(1 + i\right) 
=
4
44 
4
Rapid solution [src] 
x1 = -1 - I
x1=−1−ix_{1} = -1 - i 
x2 = -1 + I
x2=−1+ix_{2} = -1 + i 
x3 = 1 - I
x3=1−ix_{3} = 1 - i 
x4 = 1 + I
x4=1+ix_{4} = 1 + i 
x4 = 1 + i
Numerical answer [src] 
x1 = -1.0 - 1.0*i
x2 = 1.0 - 1.0*i
x3 = 1.0 + 1.0*i
x4 = -1.0 + 1.0*i
x4 = -1.0 + 1.0*i
The graph
x^4+4=0 equation
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