Solve for c 70=-10-10c^3

$70 = - 10 - 10 c^{3}$

Rewrite the equation as $- 10 - 10 c^{3} = 70$ .

$- 10 - 10 c^{3} = 70$

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

Add $10$ to both sides of the equation.

$- 10 c^{3} = 70 + 10$

Add $70$ and $10$ .

$- 10 c^{3} = 80$

Subtract $80$ from both sides of the equation.

$- 10 c^{3} - 80 = 0$

Factor the left side of the equation.

Factor $- 10$ out of $- 10 c^{3} - 80$ .

Factor $- 10$ out of $- 10 c^{3}$ .

$- 10 c^{3} - 80 = 0$

Factor $- 10$ out of $- 80$ .

$- 10 c^{3} - 10 \cdot 8 = 0$

Factor $- 10$ out of $- 10 (c^{3}) - 10 (8)$ .

$- 10 (c^{3} + 8) = 0$

Rewrite $8$ as $2^{3}$ .

$- 10 (c^{3} + 2^{3}) = 0$

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 $a = c$ and $b = 2$ .

$- 10 ((c + 2) (c^{2} - c \cdot 2 + 2^{2})) = 0$

Factor.

Simplify.

Multiply $2$ by $- 1$ .

$- 10 ((c + 2) (c^{2} - 2 c + 2^{2})) = 0$

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

$- 10 ((c + 2) (c^{2} - 2 c + 4)) = 0$

Remove unnecessary parentheses.

$- 10 (c + 2) (c^{2} - 2 c + 4) = 0$

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

$c + 2 = 0$

$c^{2} - 2 c + 4 = 0$

Set $c + 2$ equal to $0$ and solve for $c$ .

Set $c + 2$ equal to $0$ .

$c + 2 = 0$

Subtract $2$ from both sides of the equation.

$c = - 2$

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

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

$c^{2} - 2 c + 4 = 0$

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 2$ , and $c = 4$ into the quadratic formula and solve for $c$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $4$ .

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

Subtract $16$ from $4$ .

$c = \frac{2 \pm \sqrt{- 12}}{2 \cdot 1}$

Rewrite $- 12$ as $- 1 (12)$ .

$c = \frac{2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$c = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

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

$c = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

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

Pull terms out from under the radical.

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

Move $2$ to the left of $i$ .

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

Multiply $2$ by $1$ .

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

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

$c = 1 \pm i \sqrt{3}$

The final answer is the combination of both solutions.

$c = 1 + i \sqrt{3}, 1 - i \sqrt{3}$

The final solution is all the values that make $- 10 (c + 2) (c^{2} - 2 c + 4) = 0$ true.

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

Solve for c 70=-10-10c^3

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