Algebra Examples

$x^{4} = 81$

Step 1

Subtract $81$ from both sides of the equation.

$x^{4} - 81 = 0$

Step 2

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

${(x^{2})}^{2} - 81 = 0$

Step 3

Rewrite $81$ as $9^{2}$ .

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

Step 4

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 $b = 9$ .

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

Step 5

Simplify.

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Step 5.1

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

$(x^{2} + 9) (x^{2} - 3^{2}) = 0$

Step 5.2

Factor.

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Step 5.2.1

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 $b = 3$ .

$(x^{2} + 9) ((x + 3) (x - 3)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$(x^{2} + 9) (x + 3) (x - 3) = 0$

Step 6

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

$x^{2} + 9 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 7

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 7.2

Solve $x^{2} + 9 = 0$ for $x$ .

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Step 7.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 7.2.2

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

$x = \pm \sqrt{- 9}$

Step 7.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 7.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 7.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 7.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 7.2.3.4

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

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

Step 7.2.3.5

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

$x = \pm i \cdot 3$

Step 7.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 7.2.4

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

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Step 7.2.4.1

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

$x = 3 i$

Step 7.2.4.2

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

$x = - 3 i$

Step 7.2.4.3

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

$x = 3 i, - 3 i$

Step 8

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 8.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 9.2

Add $3$ to both sides of the equation.

$x = 3$

Step 10

The final solution is all the values that make $(x^{2} + 9) (x + 3) (x - 3) = 0$ true.

$x = 3 i, - 3 i, - 3, 3$