Solve by Factoring x^3+512=0

$x^{3} + 512 = 0$

Step 1

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

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

Step 2

Since both terms are perfect cubes, factor using the sum 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 3

Simplify.

Multiply $8$ by $- 1$ .

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

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

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

Step 4

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

$x + 8 = 0$

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

Step 5

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

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

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 6

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $64$ .

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

Subtract $256$ from $64$ .

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

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

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

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

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

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

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

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

Factor $64$ out of $192$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $1$ .

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

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

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $64$ .

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

Subtract $256$ from $64$ .

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

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

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

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

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

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

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

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

Factor $64$ out of $192$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $1$ .

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

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

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $64$ .

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

Subtract $256$ from $64$ .

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

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

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

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

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

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

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

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

Factor $64$ out of $192$ .

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

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

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

Pull terms out from under the radical.

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

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

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

Multiply $2$ by $1$ .

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

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

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

Step 7

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

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

Solve by Factoring x^3+512=0

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