Algebra Examples

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

Step 1

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

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

Step 2

Factor the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $3 x$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.2

Reorder terms.

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

Step 3

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

$x = 0$

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

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.2.2

Multiply $- 4$ by $3$ .

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

Step 5.2.3.1.3

Subtract $12$ from $9$ .

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

Step 5.2.3.1.4

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

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

Step 5.2.3.1.5

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

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

Step 5.2.3.1.6

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

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

Step 5.2.3.2

Multiply $2$ by $1$ .

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

Step 5.2.4

The final answer is the combination of both solutions.

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

Step 6

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

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