Algebra Examples

$x^{4} - 9 = 0$

Step 1

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

${(x^{2})}^{2} - 9 = 0$

Step 2

Rewrite $9$ as $3^{2}$ .

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 3$ .

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

Step 4

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

$x^{2} + 3 = 0$

$x^{2} - 3 = 0$

Step 5

Set $x^{2} + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 3$ equal to $0$ .

$x^{2} + 3 = 0$

Step 5.2

Solve $x^{2} + 3 = 0$ for $x$ .

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Step 5.2.1

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 5.2.2

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

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

Step 5.2.3

Simplify $\pm \sqrt{- 3}$ .

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Step 5.2.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 5.2.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 5.2.3.3

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

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

Step 5.2.4

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

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Step 5.2.4.1

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

$x = i \sqrt{3}$

Step 5.2.4.2

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

$x = - i \sqrt{3}$

Step 5.2.4.3

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

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

Step 6

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 6.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 6.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 6.2.2

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

$x = \pm \sqrt{3}$

Step 6.2.3

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

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Step 6.2.3.1

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

$x = \sqrt{3}$

Step 6.2.3.2

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

$x = - \sqrt{3}$

Step 6.2.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 7

The final solution is all the values that make $(x^{2} + 3) (x^{2} - 3) = 0$ true.

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