Find All Complex Solutions x^3+x=0

$x^{3} + x = 0$

Factor $x$ out of $x^{3} + x$ .

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} + x = 0$

Raise $x$ to the power of $1$ .

$x \cdot x^{2} + x = 0$

Factor $x$ out of $x^{1}$ .

$x \cdot x^{2} + x \cdot 1 = 0$

Factor $x$ out of $x \cdot x^{2} + x \cdot 1$ .

$x (x^{2} + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} + 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

The final solution is all the values that make $x (x^{2} + 1) = 0$ true.

$x = 0, i, - i$