Solve for w w^3-12=-76

$w^{3} - 12 = - 76$

Move all terms not containing $w$ to the right side of the equation.

Add $12$ to both sides of the equation.

$w^{3} = - 76 + 12$

Add $- 76$ and $12$ .

$w^{3} = - 64$

Add $64$ to both sides of the equation.

$w^{3} + 64 = 0$

Factor the left side of the equation.

Rewrite $64$ as $4^{3}$ .

$w^{3} + 4^{3} = 0$

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 $a = w$ and $b = 4$ .

$(w + 4) (w^{2} - w \cdot 4 + 4^{2}) = 0$

Simplify.

Multiply $4$ by $- 1$ .

$(w + 4) (w^{2} - 4 w + 4^{2}) = 0$

Raise $4$ to the power of $2$ .

$(w + 4) (w^{2} - 4 w + 16) = 0$

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

$w + 4 = 0$

$w^{2} - 4 w + 16 = 0$

Set $w + 4$ equal to $0$ and solve for $w$ .

Set $w + 4$ equal to $0$ .

$w + 4 = 0$

Subtract $4$ from both sides of the equation.

$w = - 4$

Set $w^{2} - 4 w + 16$ equal to $0$ and solve for $w$ .

Set $w^{2} - 4 w + 16$ equal to $0$ .

$w^{2} - 4 w + 16 = 0$

Solve $w^{2} - 4 w + 16 = 0$ for $w$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = 16$ into the quadratic formula and solve for $w$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$w = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$w = \frac{4 \pm \sqrt{16 - 4 \cdot 16}}{2 \cdot 1}$

Multiply $- 4$ by $16$ .

$w = \frac{4 \pm \sqrt{16 - 64}}{2 \cdot 1}$

Subtract $64$ from $16$ .

$w = \frac{4 \pm \sqrt{- 48}}{2 \cdot 1}$

Rewrite $- 48$ as $- 1 (48)$ .

$w = \frac{4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$w = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

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

$w = \frac{4 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$w = \frac{4 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$w = \frac{4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$w = \frac{4 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Move $4$ to the left of $i$ .

$w = \frac{4 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$w = \frac{4 \pm 4 i \sqrt{3}}{2}$

Simplify $\frac{4 \pm 4 i \sqrt{3}}{2}$ .

$w = 2 \pm 2 i \sqrt{3}$

The final answer is the combination of both solutions.

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

The final solution is all the values that make $(w + 4) (w^{2} - 4 w + 16) = 0$ true.

$w = - 4, 2 + 2 i \sqrt{3}, 2 - 2 i \sqrt{3}$

Solve for w w^3-12=-76

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