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9x^4-1=0

9x^4-1=0 equation

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Numerical solution:

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The solution

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   4        
9*x  - 1 = 0
9x4−1=09 x^{4} - 1 = 0
Simplify
Detail solution
Given the equation
9x4−1=09 x^{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:
94(1x+0)44=1\sqrt[4]{9} \sqrt[4]{\left(1 x + 0\right)^{4}} = 1
94(1x+0)44=−1\sqrt[4]{9} \sqrt[4]{\left(1 x + 0\right)^{4}} = -1
or
3x=1\sqrt{3} x = 1
3x=−1\sqrt{3} x = -1
Expand brackets in the left part
x*sqrt3 = 1

Divide both parts of the equation by sqrt(3)
x = 1 / (sqrt(3))

We get the answer: x = sqrt(3)/3
Expand brackets in the left part
x*sqrt3 = -1

Divide both parts of the equation by sqrt(3)
x = -1 / (sqrt(3))

We get the answer: x = -sqrt(3)/3
or
x1=−33x_{1} = - \frac{\sqrt{3}}{3}
x2=33x_{2} = \frac{\sqrt{3}}{3}

All other 2 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z4=19z^{4} = \frac{1}{9}
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r4e4ip=19r^{4} e^{4 i p} = \frac{1}{9}
where
r=33r = \frac{\sqrt{3}}{3}
- 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=−33z_{1} = - \frac{\sqrt{3}}{3}
z2=33z_{2} = \frac{\sqrt{3}}{3}
z3=−3i3z_{3} = - \frac{\sqrt{3} i}{3}
z4=3i3z_{4} = \frac{\sqrt{3} i}{3}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−33x_{1} = - \frac{\sqrt{3}}{3}
x2=33x_{2} = \frac{\sqrt{3}}{3}
x3=−3i3x_{3} = - \frac{\sqrt{3} i}{3}
x4=3i3x_{4} = \frac{\sqrt{3} i}{3}
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.515.0200-100
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
   ___      ___        ___        ___
-\/ 3     \/ 3    -I*\/ 3     I*\/ 3 
------- + ----- + --------- + -------
   3        3         3          3
(−33)+(33)+(−3i3)+(3i3)\left(- \frac{\sqrt{3}}{3}\right) + \left(\frac{\sqrt{3}}{3}\right) + \left(- \frac{\sqrt{3} i}{3}\right) + \left(\frac{\sqrt{3} i}{3}\right) 
=
0
00
Simplify 
product
   ___      ___        ___        ___
-\/ 3     \/ 3    -I*\/ 3     I*\/ 3 
------- * ----- * --------- * -------
   3        3         3          3
(−33)∗(33)∗(−3i3)∗(3i3)\left(- \frac{\sqrt{3}}{3}\right) * \left(\frac{\sqrt{3}}{3}\right) * \left(- \frac{\sqrt{3} i}{3}\right) * \left(\frac{\sqrt{3} i}{3}\right) 
=
-1/9
−19- \frac{1}{9}
Simplify
Rapid solution [src] 
         ___ 
      -\/ 3  
x_1 = -------
         3
x1=−33x_{1} = - \frac{\sqrt{3}}{3}
Simplify 
        ___
      \/ 3 
x_2 = -----
        3
x2=33x_{2} = \frac{\sqrt{3}}{3}
Simplify 
           ___ 
      -I*\/ 3  
x_3 = ---------
          3
x3=−3i3x_{3} = - \frac{\sqrt{3} i}{3}
Simplify 
          ___
      I*\/ 3 
x_4 = -------
         3
x4=3i3x_{4} = \frac{\sqrt{3} i}{3}
Simplify
Numerical answer [src] 
x1 = -0.577350269189626*i
x2 = 0.577350269189626
x3 = -0.577350269189626
x4 = 0.577350269189626*i
x4 = 0.577350269189626*i
The graph
9x^4-1=0 equation
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