Algebra Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

$x^{2} - 2 = 0$

$x^{2} + 2 = 0$

Step 2.2

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

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Step 2.2.1

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

$x^{2} - 2 = 0$

Step 2.2.2

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

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Step 2.2.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 2.2.2.2

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

$x = \pm \sqrt{2}$

Step 2.2.2.3

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

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Step 2.2.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 2.2.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 2.2.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 2.3

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

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Step 2.3.1

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

$x^{2} + 2 = 0$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 2.3.2.2

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

$x = \pm \sqrt{- 2}$

Step 2.3.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 2.3.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Step 2.3.2.3.3

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

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

Step 2.3.2.4

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

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Step 2.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 2.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 2.3.2.4.3

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

$x = i \sqrt{2}, - i \sqrt{2}$

Step 2.4

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

$x = \sqrt{2}, - \sqrt{2}, i \sqrt{2}, - i \sqrt{2}$

Step 3