Algebra Examples

x^{4} - 16 = 0

$x^{4} - 16 = 0$

Step 1

Rewrite

x^{4}

$x^{4}$ as

{(x^{2})}^{2}

${(x^{2})}^{2}$ .

{(x^{2})}^{2} - 16 = 0

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

Step 2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

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

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x^{2}

$a = x^{2}$ and

b = 4

$b = 4$ .

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

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

Step 4

Simplify.

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Step 4.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 4.2

Factor.

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Step 4.2.1

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 2

$b = 2$ .

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

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

Step 4.2.2

Remove unnecessary parentheses.

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

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

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

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

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

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

Step 5

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

0

$0$ , the entire expression will be equal to

0

$0$ .

x^{2} + 4 = 0

$x^{2} + 4 = 0$

x + 2 = 0

$x + 2 = 0$

x - 2 = 0

$x - 2 = 0$

Step 6

Set

x^{2} + 4

$x^{2} + 4$ equal to

0

$0$ and solve for

x

$x$ .

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Step 6.1

Set

x^{2} + 4

$x^{2} + 4$ equal to

0

$0$ .

x^{2} + 4 = 0

$x^{2} + 4 = 0$

Step 6.2

Solve

x^{2} + 4 = 0

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

x

$x$ .

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Step 6.2.1

Subtract

4

$4$ from both sides of the equation.

x^{2} = - 4

$x^{2} = - 4$

Step 6.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

x = \pm \sqrt{- 4}

$x = \pm \sqrt{- 4}$

Step 6.2.3

Simplify

\pm \sqrt{- 4}

$\pm \sqrt{- 4}$ .

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Step 6.2.3.1

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

x = \pm \sqrt{- 1 (4)}

$x = \pm \sqrt{- 1 (4)}$

Step 6.2.3.2

Rewrite

\sqrt{- 1 (4)}

$\sqrt{- 1 (4)}$ as

\sqrt{- 1} \cdot \sqrt{4}

$\sqrt{- 1} \cdot \sqrt{4}$ .

x = \pm \sqrt{- 1} \cdot \sqrt{4}

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

Step 6.2.3.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x = \pm i \cdot \sqrt{4}

$x = \pm i \cdot \sqrt{4}$

Step 6.2.3.4

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \pm i \cdot \sqrt{2^{2}}

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

Step 6.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

x = \pm i \cdot 2

$x = \pm i \cdot 2$

Step 6.2.3.6

Move

2

$2$ to the left of

i

$i$ .

x = \pm 2 i

$x = \pm 2 i$

x = \pm 2 i

$x = \pm 2 i$

Step 6.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.2.4.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x = 2 i

$x = 2 i$

Step 6.2.4.2

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x = - 2 i

$x = - 2 i$

Step 6.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

x = 2 i, - 2 i

$x = 2 i, - 2 i$

x = 2 i, - 2 i

$x = 2 i, - 2 i$

x = 2 i, - 2 i

$x = 2 i, - 2 i$

x = 2 i, - 2 i

$x = 2 i, - 2 i$

Step 7

Set

x + 2

$x + 2$ equal to

0

$0$ and solve for

x

$x$ .

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Step 7.1

Set

x + 2

$x + 2$ equal to

0

$0$ .

x + 2 = 0

$x + 2 = 0$

Step 7.2

Subtract

2

$2$ from both sides of the equation.

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 8

Set

x - 2

$x - 2$ equal to

0

$0$ and solve for

x

$x$ .

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Step 8.1

Set

x - 2

$x - 2$ equal to

0

$0$ .

x - 2 = 0

$x - 2 = 0$

Step 8.2

Add

2

$2$ to both sides of the equation.

x = 2

$x = 2$

x = 2

$x = 2$

Step 9

The final solution is all the values that make

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

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

x = 2 i, - 2 i, - 2, 2

$x = 2 i, - 2 i, - 2, 2$