Solve for x x^3+3x=3x^2

$x^{3} + 3 x = 3 x^{2}$

Subtract $3 x^{2}$ from both sides of the equation.

$x^{3} + 3 x - 3 x^{2} = 0$

Factor the left side of the equation.

Factor $x$ out of $x^{3} + 3 x - 3 x^{2}$ .

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

$x \cdot x^{2} + 3 x - 3 x^{2} = 0$

Factor $x$ out of $3 x$ .

$x \cdot x^{2} + x \cdot 3 - 3 x^{2} = 0$

Factor $x$ out of $- 3 x^{2}$ .

$x \cdot x^{2} + x \cdot 3 + x (- 3 x) = 0$

Factor $x$ out of $x \cdot x^{2} + x \cdot 3$ .

$x (x^{2} + 3) + x (- 3 x) = 0$

Factor $x$ out of $x (x^{2} + 3) + x (- 3 x)$ .

$x (x^{2} + 3 - 3 x) = 0$

Reorder terms.

$x (x^{2} - 3 x + 3) = 0$

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

$x = 0$

$x^{2} - 3 x + 3 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x^{2} - 3 x + 3$ equal to $0$ and solve for $x$ .

Set $x^{2} - 3 x + 3$ equal to $0$ .

$x^{2} - 3 x + 3 = 0$

Solve $x^{2} - 3 x + 3 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 3$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 3}}{2 \cdot 1}$

Multiply $- 4$ by $3$ .

$x = \frac{3 \pm \sqrt{9 - 12}}{2 \cdot 1}$

Subtract $12$ from $9$ .

$x = \frac{3 \pm \sqrt{- 3}}{2 \cdot 1}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

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

$x = \frac{3 \pm i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{3 \pm i \sqrt{3}}{2}$

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{3}}{2}, \frac{3 - i \sqrt{3}}{2}$

The final solution is all the values that make $x (x^{2} - 3 x + 3) = 0$ true.

$x = 0, \frac{3 + i \sqrt{3}}{2}, \frac{3 - i \sqrt{3}}{2}$

Solve for x x^3+3x=3x^2

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