Precalculus Examples

$x^{4} - 7 x^{2} - 8 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

${(x^{2})}^{2} - 7 x^{2} - 8 = 0$

Step 1.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 7 u - 8 = 0$

Step 1.3

Factor $u^{2} - 7 u - 8$ 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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 1.3.2

Write the factored form using these integers.

$(u - 8) (u + 1) = 0$

Step 1.4

Replace all occurrences of $u$ with $x^{2}$ .

$(x^{2} - 8) (x^{2} + 1) = 0$

Step 2

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

$x^{2} - 8 = 0$

$x^{2} + 1 = 0$

Step 3

Set $x^{2} - 8$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2} - 8$ equal to $0$ .

$x^{2} - 8 = 0$

Step 3.2

Solve $x^{2} - 8 = 0$ for $x$ .

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Step 3.2.1

Add $8$ to both sides of the equation.

$x^{2} = 8$

Step 3.2.2

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

$x = \pm \sqrt{8}$

Step 3.2.3

Simplify $\pm \sqrt{8}$ .

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Step 3.2.3.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.2.3.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 3.2.3.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.3.2

Pull terms out from under the radical.

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

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.

$x = 2 \sqrt{2}$

Step 3.2.4.2

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

$x = - 2 \sqrt{2}$

Step 3.2.4.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 4

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

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Step 4.1

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

$x^{2} + 1 = 0$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 4.2.2

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

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

Step 4.2.3

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

$x = \pm i$

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.

$x = i$

Step 4.2.4.2

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

$x = - i$

Step 4.2.4.3

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

$x = i, - i$

Step 5

The final solution is all the values that make $(x^{2} - 8) (x^{2} + 1) = 0$ true.

$x = 2 \sqrt{2}, - 2 \sqrt{2}, i, - i$