Precalculus Examples

$x^{3} - 512 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $512$ as $8^{3}$ .

$x^{3} - 8^{3} = 0$

Step 1.2

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 $b = 8$ .

$(x - 8) (x^{2} + x \cdot 8 + 8^{2}) = 0$

Step 1.3

Simplify.

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Step 1.3.1

Move $8$ to the left of $x$ .

$(x - 8) (x^{2} + 8 x + 8^{2}) = 0$

Step 1.3.2

Raise $8$ to the power of $2$ .

$(x - 8) (x^{2} + 8 x + 64) = 0$

Step 2

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

$x - 8 = 0$

$x^{2} + 8 x + 64 = 0$

Step 3

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4

Set $x^{2} + 8 x + 64$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 8 x + 64$ equal to $0$ .

$x^{2} + 8 x + 64 = 0$

Step 4.2

Solve $x^{2} + 8 x + 64 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = 8$ , and $c = 64$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot 64)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $64$ .

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $256$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 1}$

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.3.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.3.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 4.2.3.1.7.1

Factor $64$ out of $192$ .

$x = \frac{- 8 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 1}$

Step 4.2.3.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (8 \sqrt{3})}{2 \cdot 1}$

Step 4.2.3.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2}$

Step 4.2.3.3

Simplify $\frac{- 8 \pm 8 i \sqrt{3}}{2}$ .

$x = - 4 \pm 4 i \sqrt{3}$

Step 4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64}}{2 \cdot 1}$

Step 4.2.4.1.2.2

Multiply $- 4$ by $64$ .

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}$

Step 4.2.4.1.3

Subtract $256$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 1}$

Step 4.2.4.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 1}$

Step 4.2.4.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.4.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.4.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 4.2.4.1.7.1

Factor $64$ out of $192$ .

$x = \frac{- 8 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 1}$

Step 4.2.4.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (8 \sqrt{3})}{2 \cdot 1}$

Step 4.2.4.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2 \cdot 1}$

Step 4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2}$

Step 4.2.4.3

Simplify $\frac{- 8 \pm 8 i \sqrt{3}}{2}$ .

$x = - 4 \pm 4 i \sqrt{3}$

Step 4.2.4.4

Change the $\pm$ to $+$ .

$x = - 4 + 4 i \sqrt{3}$

Step 4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot 64}}{2 \cdot 1}$

Step 4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 64$ .

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Step 4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 64}}{2 \cdot 1}$

Step 4.2.5.1.2.2

Multiply $- 4$ by $64$ .

$x = \frac{- 8 \pm \sqrt{64 - 256}}{2 \cdot 1}$

Step 4.2.5.1.3

Subtract $256$ from $64$ .

$x = \frac{- 8 \pm \sqrt{- 192}}{2 \cdot 1}$

Step 4.2.5.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{- 8 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 1}$

Step 4.2.5.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.5.1.6

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

$x = \frac{- 8 \pm i \cdot \sqrt{192}}{2 \cdot 1}$

Step 4.2.5.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 4.2.5.1.7.1

Factor $64$ out of $192$ .

$x = \frac{- 8 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 1}$

Step 4.2.5.1.7.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 8 \pm i \cdot (8 \sqrt{3})}{2 \cdot 1}$

Step 4.2.5.1.9

Move $8$ to the left of $i$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2 \cdot 1}$

Step 4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 8 i \sqrt{3}}{2}$

Step 4.2.5.3

Simplify $\frac{- 8 \pm 8 i \sqrt{3}}{2}$ .

$x = - 4 \pm 4 i \sqrt{3}$

Step 4.2.5.4

Change the $\pm$ to $-$ .

$x = - 4 - 4 i \sqrt{3}$

Step 4.2.6

The final answer is the combination of both solutions.

$x = - 4 + 4 i \sqrt{3}, - 4 - 4 i \sqrt{3}$

Step 5

The final solution is all the values that make $(x - 8) (x^{2} + 8 x + 64) = 0$ true.

$x = 8, - 4 + 4 i \sqrt{3}, - 4 - 4 i \sqrt{3}$