Linear Algebra Examples

$x^{8} + 80 x^{4} - 81 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

${(x^{4})}^{2} + 80 x^{4} - 81 = 0$

Step 1.2

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

$u^{2} + 80 u - 81 = 0$

Step 1.3

Factor $u^{2} + 80 u - 81$ 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 $- 81$ and whose sum is $80$ .

$- 1, 81$

Step 1.3.2

Write the factored form using these integers.

$(u - 1) (u + 81) = 0$

Step 1.4

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

$(x^{4} - 1) (x^{4} + 81) = 0$

Step 2

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

$x^{4} - 1 = 0$

$x^{4} + 81 = 0$

Step 3

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

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Step 3.1

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

$x^{4} - 1 = 0$

Step 3.2

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

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Step 3.2.1

Add $1$ to both sides of the equation.

$x^{4} = 1$

Step 3.2.2

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

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

Step 3.2.3

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

Step 3.2.4.2

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

$x = - 1$

Step 3.2.4.3

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

$x = 1, - 1$

Step 4

Set $x^{4} + 81$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{4} + 81$ equal to $0$ .

$x^{4} + 81 = 0$

Step 4.2

Solve $x^{4} + 81 = 0$ for $x$ .

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Step 4.2.1

Subtract $81$ from both sides of the equation.

$x^{4} = - 81$

Step 4.2.2

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

$x = \pm \sqrt[4]{- 81}$

Step 4.2.3

Simplify $\pm \sqrt[4]{- 81}$ .

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Step 4.2.3.1

Rewrite $- 81$ as $3^{4} \cdot - 1$ .

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Step 4.2.3.1.1

Factor $81$ out of $- 81$ .

$x = \pm \sqrt[4]{81 (- 1)}$

Step 4.2.3.1.2

Rewrite $81$ as $3^{4}$ .

$x = \pm \sqrt[4]{3^{4} \cdot - 1}$

Step 4.2.3.2

Pull terms out from under the radical.

$x = \pm 3 \sqrt[4]{- 1}$

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.

$x = 3 \sqrt[4]{- 1}$

Step 4.2.4.2

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

$x = - 3 \sqrt[4]{- 1}$

Step 4.2.4.3

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

$x = 3 \sqrt[4]{- 1}, - 3 \sqrt[4]{- 1}$

Step 5

The final solution is all the values that make $(x^{4} - 1) (x^{4} + 81) = 0$ true.

$x = 1, - 1, 3 \sqrt[4]{- 1}, - 3 \sqrt[4]{- 1}$

Step 6

The domain is the set of all valid $x$ values.

${x | x = 1, - 1, 3 \sqrt[4]{- 1}, - 3 \sqrt[4]{- 1}}$

Step 7