Given the equation:
$$x^{3} + x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} + 1\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} + 1 = 0$$
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:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = 1$$
, 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
$$x_{2} = i$$
$$x_{3} = - i$$
The final answer for x^3 + x = 0:
$$x_{1} = 0$$
$$x_{2} = i$$
$$x_{3} = - i$$