Algebra Examples

x^{3} - 8 = 0

$x^{3} - 8 = 0$

Step 1

Rewrite

$8$ as

$2^{3}$ .

$x^{3} - 2^{3} = 0$

Step 2

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

$a = x$ and

$b = 2$ .

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

Step 3

Simplify.

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Step 3.1

Move

$2$ to the left of

$x$ .

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

Step 3.2

Raise

$2$ to the power of

$2$ .

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

Step 4

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x - 2 = 0$

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

Step 5

Set

$x - 2$ equal to

$0$ and solve for

$x$ .

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Step 5.1

Set

$x - 2$ equal to

$0$ .

$x - 2 = 0$

Step 5.2

Add

$2$ to both sides of the equation.

$x = 2$

Step 6

Set

$x^{2} + 2 x + 4$ equal to

$0$ and solve for

$x$ .

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Step 6.1

Set

$x^{2} + 2 x + 4$ equal to

$0$ .

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

Step 6.2

Solve

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

$x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values

$a = 1$ ,

$b = 2$ , and

$c = 4$ into the quadratic formula and solve for

$x$ .

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise

$2$ to the power of

$2$ .

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

Step 6.2.3.1.2

Multiply

$- 4 \cdot 1 \cdot 4$ .

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Step 6.2.3.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 6.2.3.1.2.2

Multiply

$- 4$ by

$4$ .

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

Step 6.2.3.1.3

Subtract

$16$ from

$4$ .

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

Step 6.2.3.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

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

Step 6.2.3.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 6.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

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

Step 6.2.3.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 6.2.3.1.7.1

Factor

$4$ out of

$12$ .

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

Step 6.2.3.1.7.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 6.2.3.1.8

Pull terms out from under the radical.

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

Step 6.2.3.1.9

Move

$2$ to the left of

$i$ .

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

Step 6.2.3.2

Multiply

$2$ by

$1$ .

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

Step 6.2.3.3

Simplify

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

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

Step 6.2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Raise

$2$ to the power of

$2$ .

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

Step 6.2.4.1.2

Multiply

$- 4 \cdot 1 \cdot 4$ .

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Step 6.2.4.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 6.2.4.1.2.2

Multiply

$- 4$ by

$4$ .

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

Step 6.2.4.1.3

Subtract

$16$ from

$4$ .

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

Step 6.2.4.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

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

Step 6.2.4.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 6.2.4.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

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

Step 6.2.4.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 6.2.4.1.7.1

Factor

$4$ out of

$12$ .

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

Step 6.2.4.1.7.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 6.2.4.1.8

Pull terms out from under the radical.

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

Step 6.2.4.1.9

Move

$2$ to the left of

$i$ .

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

Step 6.2.4.2

Multiply

$2$ by

$1$ .

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

Step 6.2.4.3

Simplify

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

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

Step 6.2.4.4

Change the

$\pm$ to

$+$ .

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

Step 6.2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Raise

$2$ to the power of

$2$ .

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

Step 6.2.5.1.2

Multiply

$- 4 \cdot 1 \cdot 4$ .

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Step 6.2.5.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 6.2.5.1.2.2

Multiply

$- 4$ by

$4$ .

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

Step 6.2.5.1.3

Subtract

$16$ from

$4$ .

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

Step 6.2.5.1.4

Rewrite

$- 12$ as

$- 1 (12)$ .

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

Step 6.2.5.1.5

Rewrite

$\sqrt{- 1 (12)}$ as

$\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 6.2.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

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

Step 6.2.5.1.7

Rewrite

$12$ as

$2^{2} \cdot 3$ .

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Step 6.2.5.1.7.1

Factor

$4$ out of

$12$ .

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

Step 6.2.5.1.7.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 6.2.5.1.8

Pull terms out from under the radical.

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

Step 6.2.5.1.9

Move

$2$ to the left of

$i$ .

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

Step 6.2.5.2

Multiply

$2$ by

$1$ .

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

Step 6.2.5.3

Simplify

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

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

Step 6.2.5.4

Change the

$\pm$ to

$-$ .

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

Step 6.2.6

The final answer is the combination of both solutions.

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

Step 7

The final solution is all the values that make

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

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