Algebra Examples

$x^{8} - 3 x^{4} + 2 = 0$

Step 1

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

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

Step 2

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$u^{2} - 3 u + 2 = 0$

Step 3

Factor $u^{2} - 3 u + 2$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 3.2

Write the factored form using these integers.

$(u - 2) (u - 1) = 0$

Step 4

Replace all occurrences of $u$ with $x^{4}$ .

$(x^{4} - 2) (x^{4} - 1) = 0$

Step 5

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

$(x^{4} - 2) ({(x^{2})}^{2} - 1) = 0$

Step 6

Rewrite $1$ as $1^{2}$ .

$(x^{4} - 2) ({(x^{2})}^{2} - 1^{2}) = 0$

Step 7

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 $b = 1$ .

$(x^{4} - 2) ((x^{2} + 1) (x^{2} - 1)) = 0$

Step 8

Factor.

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Step 8.1

Simplify.

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Step 8.1.1

Rewrite $1$ as $1^{2}$ .

$(x^{4} - 2) ((x^{2} + 1) (x^{2} - 1^{2})) = 0$

Step 8.1.2

Factor.

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Step 8.1.2.1

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 $b = 1$ .

$(x^{4} - 2) ((x^{2} + 1) ((x + 1) (x - 1))) = 0$

Step 8.1.2.2

Remove unnecessary parentheses.

$(x^{4} - 2) ((x^{2} + 1) (x + 1) (x - 1)) = 0$

Step 8.2

Remove unnecessary parentheses.

$(x^{4} - 2) (x^{2} + 1) (x + 1) (x - 1) = 0$

Step 9

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

$x^{4} - 2 = 0$

$x^{2} + 1 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 10

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

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Step 10.1

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

$x^{4} - 2 = 0$

Step 10.2

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

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Step 10.2.1

Add $2$ to both sides of the equation.

$x^{4} = 2$

Step 10.2.2

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

$x = \pm \sqrt[4]{2}$

Step 10.2.3

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

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Step 10.2.3.1

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

$x = \sqrt[4]{2}$

Step 10.2.3.2

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

$x = - \sqrt[4]{2}$

Step 10.2.3.3

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

$x = \sqrt[4]{2}, - \sqrt[4]{2}$

Step 11

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 11.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 11.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 11.2.2

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

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

Step 11.2.3

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

$x = \pm i$

Step 11.2.4

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

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Step 11.2.4.1

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

$x = i$

Step 11.2.4.2

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

$x = - i$

Step 11.2.4.3

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

$x = i, - i$

Step 12

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 12.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 12.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 13

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 13.2

Add $1$ to both sides of the equation.

$x = 1$

Step 14

The final solution is all the values that make $(x^{4} - 2) (x^{2} + 1) (x + 1) (x - 1) = 0$ true.

$x = \sqrt[4]{2}, - \sqrt[4]{2}, i, - i, - 1, 1$