Algebra Examples

$8 x^{3} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$8 x^{3} - 1 = 0$

Step 2

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

${(2 x)}^{3} - 1 = 0$

Step 3

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

${(2 x)}^{3} - 1^{3} = 0$

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 $a = 2 x$ and $b = 1$ .

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

Step 5

Simplify.

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Step 5.1

Apply the product rule to $2 x$ .

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

Step 5.2

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

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

Step 5.3

Multiply $2$ by $1$ .

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

Step 5.4

One to any power is one.

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

Step 6

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

$2 x - 1 = 0$

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

Step 7

Set $2 x - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 7.2

Solve $2 x - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 7.2.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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Step 8.2.1

Use the quadratic formula to find the solutions.

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

Simplify the numerator.

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Step 8.2.3.1.1

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

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

Step 8.2.3.1.2

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

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Step 8.2.3.1.2.1

Multiply $- 4$ by $4$ .

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

Step 8.2.3.1.2.2

Multiply $- 16$ by $1$ .

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

Step 8.2.3.1.3

Subtract $16$ from $4$ .

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

Step 8.2.3.1.4

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

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 8.2.3.1.5

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

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.3.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.3.1.7

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

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Step 8.2.3.1.7.1

Factor $4$ out of $12$ .

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

Step 8.2.3.1.7.2

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

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

Step 8.2.3.1.8

Pull terms out from under the radical.

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

Step 8.2.3.1.9

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

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

Step 8.2.3.2

Multiply $2$ by $4$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{8}$

Step 8.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{8}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{4}$

Step 8.2.4

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

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Step 8.2.4.1

Simplify the numerator.

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Step 8.2.4.1.1

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

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

Step 8.2.4.1.2

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

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Step 8.2.4.1.2.1

Multiply $- 4$ by $4$ .

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

Step 8.2.4.1.2.2

Multiply $- 16$ by $1$ .

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

Step 8.2.4.1.3

Subtract $16$ from $4$ .

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

Step 8.2.4.1.4

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

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 8.2.4.1.5

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

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.4.1.7

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

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Step 8.2.4.1.7.1

Factor $4$ out of $12$ .

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

Step 8.2.4.1.7.2

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

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

Step 8.2.4.1.8

Pull terms out from under the radical.

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

Step 8.2.4.1.9

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

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

Step 8.2.4.2

Multiply $2$ by $4$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{8}$

Step 8.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{8}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{4}$

Step 8.2.4.4

Change the $\pm$ to $+$ .

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

Step 8.2.4.5

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

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

Step 8.2.4.6

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

$x = \frac{- 1 \cdot 1 - (- i \sqrt{3})}{4}$

Step 8.2.4.7

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

$x = \frac{- 1 (1 - i \sqrt{3})}{4}$

Step 8.2.4.8

Move the negative in front of the fraction.

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

Step 8.2.5

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

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Step 8.2.5.1

Simplify the numerator.

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Step 8.2.5.1.1

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

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

Step 8.2.5.1.2

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

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Step 8.2.5.1.2.1

Multiply $- 4$ by $4$ .

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

Step 8.2.5.1.2.2

Multiply $- 16$ by $1$ .

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

Step 8.2.5.1.3

Subtract $16$ from $4$ .

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

Step 8.2.5.1.4

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

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 4}$

Step 8.2.5.1.5

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

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.5.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 4}$

Step 8.2.5.1.7

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

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Step 8.2.5.1.7.1

Factor $4$ out of $12$ .

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

Step 8.2.5.1.7.2

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

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

Step 8.2.5.1.8

Pull terms out from under the radical.

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

Step 8.2.5.1.9

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

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

Step 8.2.5.2

Multiply $2$ by $4$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{8}$

Step 8.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{8}$ .

$x = \frac{- 1 \pm i \sqrt{3}}{4}$

Step 8.2.5.4

Change the $\pm$ to $-$ .

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

Step 8.2.5.5

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

$x = \frac{- 1 \cdot 1 - i \sqrt{3}}{4}$

Step 8.2.5.6

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

$x = \frac{- 1 \cdot 1 - (i \sqrt{3})}{4}$

Step 8.2.5.7

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

$x = \frac{- 1 (1 + i \sqrt{3})}{4}$

Step 8.2.5.8

Move the negative in front of the fraction.

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

Step 8.2.6

The final answer is the combination of both solutions.

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

Step 9

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

$x = \frac{1}{2}, - \frac{1 - i \sqrt{3}}{4}, - \frac{1 + i \sqrt{3}}{4}$