Solve by Factoring 81x^3-192=0

$81 x^{3} - 192 = 0$

Step 1

Factor $3$ out of $81 x^{3} - 192$ .

Factor $3$ out of $81 x^{3}$ .

$3 (27 x^{3}) - 192 = 0$

Factor $3$ out of $- 192$ .

$3 (27 x^{3}) + 3 (- 64) = 0$

Factor $3$ out of $3 (27 x^{3}) + 3 (- 64)$ .

$3 (27 x^{3} - 64) = 0$

Step 2

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

$3 ({(3 x)}^{3} - 64) = 0$

Step 3

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

$3 ({(3 x)}^{3} - 4^{3}) = 0$

Step 4

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 $a = 3 x$ and $b = 4$ .

$3 ((3 x - 4) ({(3 x)}^{2} + 3 x \cdot 4 + 4^{2})) = 0$

Step 5

Factor.

Simplify.

Apply the product rule to $3 x$ .

$3 ((3 x - 4) (3^{2} x^{2} + 3 x \cdot 4 + 4^{2})) = 0$

Raise $3$ to the power of $2$ .

$3 ((3 x - 4) (9 x^{2} + 3 x \cdot 4 + 4^{2})) = 0$

Multiply $4$ by $3$ .

$3 ((3 x - 4) (9 x^{2} + 12 x + 4^{2})) = 0$

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

$3 ((3 x - 4) (9 x^{2} + 12 x + 16)) = 0$

Remove unnecessary parentheses.

$3 (3 x - 4) (9 x^{2} + 12 x + 16) = 0$

Step 6

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

$3 x - 4 = 0$

$9 x^{2} + 12 x + 16 = 0$

Step 7

Set $3 x - 4$ equal to $0$ and solve for $x$ .

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Solve $3 x - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

$3 x = 4$

Divide each term in $3 x = 4$ by $3$ and simplify.

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 8

Set $9 x^{2} + 12 x + 16$ equal to $0$ and solve for $x$ .

Set $9 x^{2} + 12 x + 16$ equal to $0$ .

$9 x^{2} + 12 x + 16 = 0$

Solve $9 x^{2} + 12 x + 16 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 9$ , $b = 12$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (9 \cdot 16)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot 16}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 16$ .

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot 16}}{2 \cdot 9}$

Multiply $- 36$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 9}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 9}$

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 9}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 9}$

Rewrite $432$ as $12^{2} \cdot 3$ .

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 9}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 9}$

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{18}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{18}$ .

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

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot 16}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 16$ .

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot 16}}{2 \cdot 9}$

Multiply $- 36$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 9}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 9}$

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 9}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 9}$

Rewrite $432$ as $12^{2} \cdot 3$ .

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 9}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 9}$

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{18}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{18}$ .

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

Change the $\pm$ to $+$ .

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

Rewrite $- 2$ as $- 1 (2)$ .

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

Factor $- 1$ out of $2 i \sqrt{3}$ .

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

Factor $- 1$ out of $- 1 (2) - (- 2 i \sqrt{3})$ .

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 9 \cdot 16}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 16$ .

Multiply $- 4$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 36 \cdot 16}}{2 \cdot 9}$

Multiply $- 36$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 9}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 9}$

Rewrite $- 432$ as $- 1 (432)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (432)}$ as $\sqrt{- 1} \cdot \sqrt{432}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 9}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 9}$

Rewrite $432$ as $12^{2} \cdot 3$ .

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 9}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 9}$

Move $12$ to the left of $i$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{18}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{18}$ .

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

Change the $\pm$ to $-$ .

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

Rewrite $- 2$ as $- 1 (2)$ .

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

Factor $- 1$ out of $- 2 i \sqrt{3}$ .

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

Factor $- 1$ out of $- 1 (2) - (2 i \sqrt{3})$ .

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

Move the negative in front of the fraction.

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

The final answer is the combination of both solutions.

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

Step 9

The final solution is all the values that make $3 (3 x - 4) (9 x^{2} + 12 x + 16) = 0$ true.

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

Solve by Factoring 81x^3-192=0

Related Questions

Support LCMGCF.com by disabling your adblocker.