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