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x^3=27

x^3=27 equation

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Numerical solution:

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The solution

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 3     
x  = 27
x3=27x^{3} = 27
Simplify
Detail solution
Given the equation
x3=27x^{3} = 27
Because equation degree is equal to = 3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root 3-th degree of the equation sides:
We get:
x33=273\sqrt[3]{x^{3}} = \sqrt[3]{27}
or
x=3x = 3
We get the answer: x = 3

All other 2 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z3=27z^{3} = 27
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r3e3ip=27r^{3} e^{3 i p} = 27
where
r=3r = 3
- the magnitude of the complex number
Substitute r:
e3ip=1e^{3 i p} = 1
Using Euler’s formula, we find roots for p
isin⁡(3p)+cos⁡(3p)=1i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = 1
so
cos⁡(3p)=1\cos{\left(3 p \right)} = 1
and
sin⁡(3p)=0\sin{\left(3 p \right)} = 0
then
p=2πN3p = \frac{2 \pi N}{3}
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=3z_{1} = 3
z2=−32−33i2z_{2} = - \frac{3}{2} - \frac{3 \sqrt{3} i}{2}
z3=−32+33i2z_{3} = - \frac{3}{2} + \frac{3 \sqrt{3} i}{2}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=3x_{1} = 3
x2=−32−33i2x_{2} = - \frac{3}{2} - \frac{3 \sqrt{3} i}{2}
x3=−32+33i2x_{3} = - \frac{3}{2} + \frac{3 \sqrt{3} i}{2}
Vieta's Theorem
it is reduced cubic equation
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−27v = -27
Vieta Formulas
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=0x_{1} + x_{2} + x_{3} = 0
x1x2+x1x3+x2x3=0x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0
x1x2x3=−27x_{1} x_{2} x_{3} = -27
The graph
-10.0-7.5-5.0-2.50.02.55.07.510.012.515.017.5-25002500
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = 3
x1=3x_{1} = 3 
                 ___
       3   3*I*\/ 3 
x2 = - - - ---------
       2       2
x2=−32−33i2x_{2} = - \frac{3}{2} - \frac{3 \sqrt{3} i}{2} 
                 ___
       3   3*I*\/ 3 
x3 = - - + ---------
       2       2
x3=−32+33i2x_{3} = - \frac{3}{2} + \frac{3 \sqrt{3} i}{2} 
x3 = -3/2 + 3*sqrt(3)*i/2
Sum and product of roots [src] 
sum
                ___               ___
      3   3*I*\/ 3      3   3*I*\/ 3 
3 + - - - --------- + - - + ---------
      2       2         2       2
(3+(−32−33i2))+(−32+33i2)\left(3 + \left(- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}\right)\right) + \left(- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}\right) 
=
0
00 
product
  /            ___\ /            ___\
  |  3   3*I*\/ 3 | |  3   3*I*\/ 3 |
3*|- - - ---------|*|- - + ---------|
  \  2       2    / \  2       2    /
3(−32−33i2)(−32+33i2)3 \left(- \frac{3}{2} - \frac{3 \sqrt{3} i}{2}\right) \left(- \frac{3}{2} + \frac{3 \sqrt{3} i}{2}\right) 
=
27
2727 
27
Numerical answer [src] 
x1 = -1.5 + 2.59807621135332*i
x2 = -1.5 - 2.59807621135332*i
x3 = 3.0
x3 = 3.0
The graph
x^3=27 equation
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