Calculus Examples

$x (x^{3} - 1) = 0$

Step 1

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

$x = 0$

$x^{3} - 1 = 0$

Step 2

Set $x$ equal to $0$ .

$x = 0$

Step 3

Set $x^{3} - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{3} - 1$ equal to $0$ .

$x^{3} - 1 = 0$

Step 3.2

Solve $x^{3} - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$x^{3} = 1$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$x^{3} - 1 = 0$

Step 3.2.3

Factor the left side of the equation.

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Step 3.2.3.1

Rewrite $1$ as $1^{3}$ .

$x^{3} - 1^{3} = 0$

Step 3.2.3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 3.2.3.3

Simplify.

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Step 3.2.3.3.1

Multiply $x$ by $1$ .

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

Step 3.2.3.3.2

One to any power is one.

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

Step 3.2.4

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

$x - 1 = 0$

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

Step 3.2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.2.6

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

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Step 3.2.6.1

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

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

Step 3.2.6.2

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

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Step 3.2.6.2.1

Use the quadratic formula to find the solutions.

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

Step 3.2.6.2.2

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

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

Step 3.2.6.2.3

Simplify.

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Step 3.2.6.2.3.1

Simplify the numerator.

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Step 3.2.6.2.3.1.1

One to any power is one.

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

Step 3.2.6.2.3.1.2

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

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Step 3.2.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.2.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 3.2.6.2.3.1.3

Subtract $4$ from $1$ .

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

Step 3.2.6.2.3.1.4

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

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

Step 3.2.6.2.3.1.5

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

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

Step 3.2.6.2.3.1.6

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

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

Step 3.2.6.2.3.2

Multiply $2$ by $1$ .

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

Step 3.2.6.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 3.2.7

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

$x = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 4

The final solution is all the values that make $x (x^{3} - 1) = 0$ true.

$x = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$