x³+x=0

To solve the equation \( x^3 + x = 0 \), we can start by factoring out the common term \( x \):

\[ x(x^2 + 1) = 0 \]

Now, we set each factor equal to zero:

1. \( x = 0 \)
2. \( x^2 + 1 = 0 \)

For the second equation, \( x^2 + 1 = 0 \), we can solve for \( x^2 \):

\[ x^2 = -1 \]

Taking the square root of both sides gives us:

\[ x = i \quad \text{or} \quad x = -i \]

where \( i \) is the imaginary unit.

Therefore, the solutions to the equation \( x^3 + x = 0 \) are:

\[ x = 0, \quad x = i, \quad x = -i \]