Solve by Factoring 64x^3-27=0

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

Step 1

Rewrite $64 x^{3}$ as ${(4 x)}^{3}$ .

${(4 x)}^{3} - 27 = 0$

Step 2

Rewrite $27$ as $3^{3}$ .

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

Step 3

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

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

Step 4

Simplify.

Apply the product rule to $4 x$ .

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

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

$(4 x - 3) (16 x^{2} + 4 x \cdot 3 + 3^{2}) = 0$

Multiply $3$ by $4$ .

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

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

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

Step 5

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

$4 x - 3 = 0$

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

Step 6

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

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

$4 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$4 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 7

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

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

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

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

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

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

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

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

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

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

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

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

Change the $\pm$ to $+$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

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

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64 \cdot 9}}{2 \cdot 16}$

Multiply $- 64$ by $9$ .

$x = \frac{- 12 \pm \sqrt{144 - 576}}{2 \cdot 16}$

Subtract $576$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{432}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{432}}{2 \cdot 16}$

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

Factor $144$ out of $432$ .

$x = \frac{- 12 \pm i \cdot \sqrt{144 (3)}}{2 \cdot 16}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{12^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (12 \sqrt{3})}{2 \cdot 16}$

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

$x = \frac{- 12 \pm 12 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 12 \pm 12 i \sqrt{3}}{32}$

Simplify $\frac{- 12 \pm 12 i \sqrt{3}}{32}$ .

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

Change the $\pm$ to $-$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

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

$x = \frac{- 1 (3 + 3 i \sqrt{3})}{8}$

Move the negative in front of the fraction.

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

The final answer is the combination of both solutions.

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

Step 8

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

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

Solve by Factoring 64x^3-27=0

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