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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

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 4        
x  - 4 = 0
$$x^{4} - 4 = 0$$
Detail solution
Given the equation
$$x^{4} - 4 = 0$$
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:
$$\sqrt[4]{x^{4}} = \sqrt[4]{4}$$
$$\sqrt[4]{x^{4}} = \left(-1\right) \sqrt[4]{4}$$
or
$$x = \sqrt{2}$$
$$x = - \sqrt{2}$$
Expand brackets in the right part
x = sqrt2

We get the answer: x = sqrt(2)
Expand brackets in the right part
x = -sqrt2

We get the answer: x = -sqrt(2)
or
$$x_{1} = - \sqrt{2}$$
$$x_{2} = \sqrt{2}$$

All other 2 root(s) is the complex numbers.
do replacement:
$$z = x$$
then the equation will be the:
$$z^{4} = 4$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 4$$
where
$$r = \sqrt{2}$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = 1$$
so
$$\cos{\left(4 p \right)} = 1$$
and
$$\sin{\left(4 p \right)} = 0$$
then
$$p = \frac{\pi N}{2}$$
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:
$$z_{1} = - \sqrt{2}$$
$$z_{2} = \sqrt{2}$$
$$z_{3} = - \sqrt{2} i$$
$$z_{4} = \sqrt{2} i$$
do backward replacement
$$z = x$$
$$x = z$$

The final answer:
$$x_{1} = - \sqrt{2}$$
$$x_{2} = \sqrt{2}$$
$$x_{3} = - \sqrt{2} i$$
$$x_{4} = \sqrt{2} i$$
The graph
Sum and product of roots [src]
sum
    ___     ___       ___       ___
- \/ 2  + \/ 2  - I*\/ 2  + I*\/ 2 
$$\left(\left(- \sqrt{2} + \sqrt{2}\right) - \sqrt{2} i\right) + \sqrt{2} i$$
=
0
$$0$$
product
   ___   ___ /     ___\     ___
-\/ 2 *\/ 2 *\-I*\/ 2 /*I*\/ 2 
$$\sqrt{2} i - \sqrt{2} \sqrt{2} \left(- \sqrt{2} i\right)$$
=
-4
$$-4$$
-4
Rapid solution [src]
        ___
x1 = -\/ 2 
$$x_{1} = - \sqrt{2}$$
       ___
x2 = \/ 2 
$$x_{2} = \sqrt{2}$$
          ___
x3 = -I*\/ 2 
$$x_{3} = - \sqrt{2} i$$
         ___
x4 = I*\/ 2 
$$x_{4} = \sqrt{2} i$$
x4 = sqrt(2)*i
Numerical answer [src]
x1 = -1.4142135623731*i
x2 = 1.4142135623731
x3 = 1.4142135623731*i
x4 = -1.4142135623731
x4 = -1.4142135623731
The graph
x^4-4=0 equation