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x^3+x=0

x^3+x=0 equation

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Numerical solution:

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The solution

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x  + x = 0
x3+x=0x^{3} + x = 0
Simplify
Detail solution
Given the equation:
x3+x=0x^{3} + x = 0
transform
Take common factor x from the equation
we get:
x(x2+1)=0x \left(x^{2} + 1\right) = 0
then:
x1=0x_{1} = 0
and also
we get the equation
x2+1=0x^{2} + 1 = 0
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
x2=D−b2ax_{2} = \frac{\sqrt{D} - b}{2 a}
x3=−D−b2ax_{3} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=0b = 0
c=1c = 1
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (1) = -4

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)

x3 = (-b - sqrt(D)) / (2*a)

or
x2=ix_{2} = i
x3=−ix_{3} = - i
The final answer for x^3 + x = 0:
x1=0x_{1} = 0
x2=ix_{2} = i
x3=−ix_{3} = - i
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=1q = 1
v=dav = \frac{d}{a}
v=0v = 0
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=1x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1
x1x2x3=0x_{1} x_{2} x_{3} = 0
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.515.010.012.5-20002000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
-I + I
−i+i- i + i 
=
0
00 
product
0*(-I)*I
i0(−i)i 0 \left(- i\right) 
=
0
00 
0
Rapid solution [src] 
x1 = 0
x1=0x_{1} = 0 
x2 = -I
x2=−ix_{2} = - i 
x3 = I
x3=ix_{3} = i 
x3 = i
Numerical answer [src] 
x1 = 0.0
x2 = 1.0*i
x3 = -1.0*i
x3 = -1.0*i
The graph
x^3+x=0 equation
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