Algebra Examples

$x^{8} - 26 x^{4} + 25 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

${(x^{4})}^{2} - 26 x^{4} + 25 = 0$

Step 1.2

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

$u^{2} - 26 u + 25 = 0$

Step 1.3

Factor $u^{2} - 26 u + 25$ using the AC method.

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Step 1.3.1

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 $25$ and whose sum is $- 26$ .

$- 25, - 1$

Step 1.3.2

Write the factored form using these integers.

$(u - 25) (u - 1) = 0$

Step 1.4

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

$(x^{4} - 25) (x^{4} - 1) = 0$

Step 2

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

$x^{4} - 25 = 0$

$x^{4} - 1 = 0$

Step 3

Set $x^{4} - 25$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{4} - 25$ equal to $0$ .

$x^{4} - 25 = 0$

Step 3.2

Solve $x^{4} - 25 = 0$ for $x$ .

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Step 3.2.1

Add $25$ to both sides of the equation.

$x^{4} = 25$

Step 3.2.2

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

$x = \pm \sqrt[4]{25}$

Step 3.2.3

Simplify $\pm \sqrt[4]{25}$ .

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Step 3.2.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt[4]{5^{2}}$

Step 3.2.3.2

Rewrite $\sqrt[4]{5^{2}}$ as $\sqrt{\sqrt{5^{2}}}$ .

$x = \pm \sqrt{\sqrt{5^{2}}}$

Step 3.2.3.3

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

$x = \pm \sqrt{5}$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = \sqrt{5}$

Step 3.2.4.2

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

$x = - \sqrt{5}$

Step 3.2.4.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 4

Set $x^{4} - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{4} - 1$ equal to $0$ .

$x^{4} - 1 = 0$

Step 4.2

Solve $x^{4} - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$x^{4} = 1$

Step 4.2.2

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

$x = \pm \sqrt[4]{1}$

Step 4.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.2.4

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

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Step 4.2.4.1

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

$x = 1$

Step 4.2.4.2

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

$x = - 1$

Step 4.2.4.3

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

$x = 1, - 1$

Step 5

The final solution is all the values that make $(x^{4} - 25) (x^{4} - 1) = 0$ true.

$x = \sqrt{5}, - \sqrt{5}, 1, - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{5}, - \sqrt{5}, 1, - 1$

Decimal Form:

$x = 2.23606797 \dots, - 2.23606797 \dots, 1, - 1$

Step 7