Algebra Examples

$x^{6} - 64 = 0$

Step 1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

${(x^{2})}^{3} - 64 = 0$

Step 2

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

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

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 $a = x^{2}$ and $b = 4$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 4.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.3.2

Multiply $2$ by $2$ .

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

Step 4.4

Move $4$ to the left of $x^{2}$ .

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

Step 4.5

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

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

Step 4.6

Factor.

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Step 4.6.1

Rewrite $x^{4} + 4 x^{2} + 16$ in a factored form.

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Step 4.6.1.1

Rewrite the middle term.

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

Step 4.6.1.2

Rearrange terms.

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

Step 4.6.1.3

Factor first three terms by perfect square rule.

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

Step 4.6.1.4

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 4.6.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2} + 4$ and $b = 2 x$ .

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

Step 4.6.1.6

Simplify.

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Step 4.6.1.6.1

Reorder terms.

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

Step 4.6.1.6.2

Multiply $2$ by $- 1$ .

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

Step 4.6.1.6.3

Reorder terms.

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

Step 4.6.2

Remove unnecessary parentheses.

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

Step 5

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

$x + 2 = 0$

$x - 2 = 0$

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

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

Step 6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7

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

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Step 7.1

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

$x - 2 = 0$

Step 7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 8

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

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Step 8.1

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

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

Step 8.2

Solve $x^{2} + 2 x + 4 = 0$ 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 $a = 1$ , $b = 2$ , 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 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 $2$ .

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

Step 8.2.3.1.2

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

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Step 8.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.2.3.1.2.2

Multiply $- 4$ by $4$ .

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

Step 8.2.3.1.3

Subtract $16$ from $4$ .

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

Step 8.2.3.1.4

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

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

Step 8.2.3.1.5

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

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

Step 8.2.3.1.6

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

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

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 $12$ .

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

Step 8.2.3.1.7.2

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

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

Step 8.2.3.1.8

Pull terms out from under the radical.

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

Step 8.2.3.1.9

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

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

Step 8.2.3.2

Multiply $2$ by $1$ .

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

Step 8.2.3.3

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

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

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 $2$ .

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

Step 8.2.4.1.2

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

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Step 8.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.2.4.1.2.2

Multiply $- 4$ by $4$ .

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

Step 8.2.4.1.3

Subtract $16$ from $4$ .

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

Step 8.2.4.1.4

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

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

Step 8.2.4.1.5

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

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

Step 8.2.4.1.6

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

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

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 $12$ .

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

Step 8.2.4.1.7.2

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

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

Step 8.2.4.1.8

Pull terms out from under the radical.

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

Step 8.2.4.1.9

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

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

Step 8.2.4.2

Multiply $2$ by $1$ .

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

Step 8.2.4.3

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

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

Step 8.2.4.4

Change the $\pm$ to $+$ .

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

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 $2$ .

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

Step 8.2.5.1.2

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

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Step 8.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.2.5.1.2.2

Multiply $- 4$ by $4$ .

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

Step 8.2.5.1.3

Subtract $16$ from $4$ .

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

Step 8.2.5.1.4

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

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

Step 8.2.5.1.5

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

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

Step 8.2.5.1.6

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

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

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 $12$ .

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

Step 8.2.5.1.7.2

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

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

Step 8.2.5.1.8

Pull terms out from under the radical.

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

Step 8.2.5.1.9

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

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

Step 8.2.5.2

Multiply $2$ by $1$ .

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

Step 8.2.5.3

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

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

Step 8.2.5.4

Change the $\pm$ to $-$ .

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

Step 8.2.6

The final answer is the combination of both solutions.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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Step 9.2.1

Use the quadratic formula to find the solutions.

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

Step 9.2.2

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 9.2.3

Simplify.

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Step 9.2.3.1

Simplify the numerator.

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Step 9.2.3.1.1

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

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

Step 9.2.3.1.2

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

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Step 9.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.2.3.1.2.2

Multiply $- 4$ by $4$ .

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

Step 9.2.3.1.3

Subtract $16$ from $4$ .

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

Step 9.2.3.1.4

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

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

Step 9.2.3.1.5

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

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

Step 9.2.3.1.6

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

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

Step 9.2.3.1.7

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

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Step 9.2.3.1.7.1

Factor $4$ out of $12$ .

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

Step 9.2.3.1.7.2

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

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

Step 9.2.3.1.8

Pull terms out from under the radical.

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

Step 9.2.3.1.9

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

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

Step 9.2.3.2

Multiply $2$ by $1$ .

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

Step 9.2.3.3

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

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

Step 9.2.4

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

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Step 9.2.4.1

Simplify the numerator.

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Step 9.2.4.1.1

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

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

Step 9.2.4.1.2

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

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Step 9.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.2.4.1.2.2

Multiply $- 4$ by $4$ .

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

Step 9.2.4.1.3

Subtract $16$ from $4$ .

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

Step 9.2.4.1.4

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

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

Step 9.2.4.1.5

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

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

Step 9.2.4.1.6

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

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

Step 9.2.4.1.7

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

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Step 9.2.4.1.7.1

Factor $4$ out of $12$ .

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

Step 9.2.4.1.7.2

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

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

Step 9.2.4.1.8

Pull terms out from under the radical.

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

Step 9.2.4.1.9

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

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

Step 9.2.4.2

Multiply $2$ by $1$ .

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

Step 9.2.4.3

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

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

Step 9.2.4.4

Change the $\pm$ to $+$ .

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

Step 9.2.5

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

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Step 9.2.5.1

Simplify the numerator.

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Step 9.2.5.1.1

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

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

Step 9.2.5.1.2

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

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Step 9.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.2.5.1.2.2

Multiply $- 4$ by $4$ .

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

Step 9.2.5.1.3

Subtract $16$ from $4$ .

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

Step 9.2.5.1.4

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

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

Step 9.2.5.1.5

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

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

Step 9.2.5.1.6

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

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

Step 9.2.5.1.7

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

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Step 9.2.5.1.7.1

Factor $4$ out of $12$ .

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

Step 9.2.5.1.7.2

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

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

Step 9.2.5.1.8

Pull terms out from under the radical.

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

Step 9.2.5.1.9

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

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

Step 9.2.5.2

Multiply $2$ by $1$ .

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

Step 9.2.5.3

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

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

Step 9.2.5.4

Change the $\pm$ to $-$ .

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

Step 9.2.6

The final answer is the combination of both solutions.

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

Step 10

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

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