Solve by Factoring 64x^3-1=0

$64 x^{3} - 1 = 0$

Step 1

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

${(4 x)}^{3} - 1 = 0$

Step 2

Rewrite $1$ as $1^{3}$ .

${(4 x)}^{3} - 1^{3} = 0$

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 $b = 1$ .

$(4 x - 1) ({(4 x)}^{2} + 4 x \cdot 1 + 1^{2}) = 0$

Step 4

Simplify.

Apply the product rule to $4 x$ .

$(4 x - 1) (4^{2} x^{2} + 4 x \cdot 1 + 1^{2}) = 0$

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

$(4 x - 1) (16 x^{2} + 4 x \cdot 1 + 1^{2}) = 0$

Multiply $4$ by $1$ .

$(4 x - 1) (16 x^{2} + 4 x + 1^{2}) = 0$

One to any power is one.

$(4 x - 1) (16 x^{2} + 4 x + 1) = 0$

Step 5

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

$4 x - 1 = 0$

$16 x^{2} + 4 x + 1 = 0$

Step 6

Set $4 x - 1$ equal to $0$ and solve for $x$ .

Set $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Solve $4 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 7

Set $16 x^{2} + 4 x + 1$ equal to $0$ and solve for $x$ .

Set $16 x^{2} + 4 x + 1$ equal to $0$ .

$16 x^{2} + 4 x + 1 = 0$

Solve $16 x^{2} + 4 x + 1 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 16 \cdot 1$ .

Multiply $- 4$ by $16$ .

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

Multiply $- 64$ by $1$ .

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

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

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

Factor $16$ out of $48$ .

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

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

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

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

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

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

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

Simplify the numerator.

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

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

Multiply $- 4 \cdot 16 \cdot 1$ .

Multiply $- 4$ by $16$ .

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

Multiply $- 64$ by $1$ .

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

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

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

Factor $16$ out of $48$ .

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

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

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

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

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

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

Change the $\pm$ to $+$ .

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

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

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

Multiply $- 4 \cdot 16 \cdot 1$ .

Multiply $- 4$ by $16$ .

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

Multiply $- 64$ by $1$ .

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

Subtract $64$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 16}$

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

$x = \frac{- 4 \pm i \cdot \sqrt{48}}{2 \cdot 16}$

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

Factor $16$ out of $48$ .

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

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

$x = \frac{- 4 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 16}$

Pull terms out from under the radical.

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

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

$x = \frac{- 4 \pm 4 i \sqrt{3}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{- 4 \pm 4 i \sqrt{3}}{32}$

Simplify $\frac{- 4 \pm 4 i \sqrt{3}}{32}$ .

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

Change the $\pm$ to $-$ .

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

The final answer is the combination of both solutions.

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

Step 8

The final solution is all the values that make $(4 x - 1) (16 x^{2} + 4 x + 1) = 0$ true.

$x = \frac{1}{4}, - \frac{1 - i \sqrt{3}}{8}, - \frac{1 + i \sqrt{3}}{8}$

Solve by Factoring 64x^3-1=0

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