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x^3=125

x^3=125 equation

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Numerical solution:

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The solution

You have entered [src] 
 3      
x  = 125
x3=125x^{3} = 125
Simplify
Detail solution
Given the equation
x3=125x^{3} = 125
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:
(1x+0)33=1253\sqrt[3]{\left(1 x + 0\right)^{3}} = \sqrt[3]{125}
or
x=5x = 5
We get the answer: x = 5

All other 2 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
z3=125z^{3} = 125
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r3e3ip=125r^{3} e^{3 i p} = 125
where
r=5r = 5
- 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=5z_{1} = 5
z2=−52−53i2z_{2} = - \frac{5}{2} - \frac{5 \sqrt{3} i}{2}
z3=−52+53i2z_{3} = - \frac{5}{2} + \frac{5 \sqrt{3} i}{2}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=5x_{1} = 5
x2=−52−53i2x_{2} = - \frac{5}{2} - \frac{5 \sqrt{3} i}{2}
x3=−52+53i2x_{3} = - \frac{5}{2} + \frac{5 \sqrt{3} i}{2}
Vieta's Theorem
it is reduced cubic equation
px2+x3+qx+v=0p x^{2} + x^{3} + q x + v = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−125v = -125
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=−125x_{1} x_{2} x_{3} = -125
The graph
-7.5-5.0-2.50.02.55.07.522.510.012.515.017.520.0-250250
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x_1 = 5
x1=5x_{1} = 5
Simplify 
                  ___
        5   5*I*\/ 3 
x_2 = - - - ---------
        2       2
x2=−52−53i2x_{2} = - \frac{5}{2} - \frac{5 \sqrt{3} i}{2}
Simplify 
                  ___
        5   5*I*\/ 3 
x_3 = - - + ---------
        2       2
x3=−52+53i2x_{3} = - \frac{5}{2} + \frac{5 \sqrt{3} i}{2}
Simplify
Sum and product of roots [src] 
sum
                ___               ___
      5   5*I*\/ 3      5   5*I*\/ 3 
5 + - - - --------- + - - + ---------
      2       2         2       2
(5)+(−52−53i2)+(−52+53i2)\left(5\right) + \left(- \frac{5}{2} - \frac{5 \sqrt{3} i}{2}\right) + \left(- \frac{5}{2} + \frac{5 \sqrt{3} i}{2}\right) 
=
0
00
Simplify 
product
                ___               ___
      5   5*I*\/ 3      5   5*I*\/ 3 
5 * - - - --------- * - - + ---------
      2       2         2       2
(5)∗(−52−53i2)∗(−52+53i2)\left(5\right) * \left(- \frac{5}{2} - \frac{5 \sqrt{3} i}{2}\right) * \left(- \frac{5}{2} + \frac{5 \sqrt{3} i}{2}\right) 
=
125
125125
Simplify
Numerical answer [src] 
x1 = 5.0
x2 = -2.5 - 4.33012701892219*i
x3 = -2.5 + 4.33012701892219*i
x3 = -2.5 + 4.33012701892219*i
The graph
x^3=125 equation
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