Precalculus Examples

$x^{4} - 80 x^{2} + 1024 < 0$

Step 1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 80 u + 1024 = 0$

$u = x^{2}$

Step 2

Factor $u^{2} - 80 u + 1024$ using the AC method.

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Step 2.1

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 $1024$ and whose sum is $- 80$ .

$- 64, - 16$

Step 2.2

Write the factored form using these integers.

$(u - 64) (u - 16) = 0$

Step 3

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

$u - 64 = 0$

$u - 16 = 0$

Step 4

Set $u - 64$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $u - 64$ equal to $0$ .

$u - 64 = 0$

Step 4.2

Add $64$ to both sides of the equation.

$u = 64$

Step 5

Set $u - 16$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 16$ equal to $0$ .

$u - 16 = 0$

Step 5.2

Add $16$ to both sides of the equation.

$u = 16$

Step 6

The final solution is all the values that make $(u - 64) (u - 16) = 0$ true.

$u = 64, 16$

Step 7

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 64$

${(x^{2})}^{1} = 16$

Step 8

Solve the first equation for $x$ .

$x^{2} = 64$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x = \pm \sqrt{64}$

Step 9.2

Simplify $\pm \sqrt{64}$ .

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Step 9.2.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 9.2.2

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

$x = \pm 8$

Step 9.3

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

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Step 9.3.1

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

$x = 8$

Step 9.3.2

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

$x = - 8$

Step 9.3.3

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

$x = 8, - 8$

Step 10

Solve the second equation for $x$ .

${(x^{2})}^{1} = 16$

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = 16$

Step 11.2

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

$x = \pm \sqrt{16}$

Step 11.3

Simplify $\pm \sqrt{16}$ .

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Step 11.3.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 11.3.2

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

$x = \pm 4$

Step 11.4

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

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Step 11.4.1

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

$x = 4$

Step 11.4.2

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

$x = - 4$

Step 11.4.3

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

$x = 4, - 4$

Step 12

The solution to $x^{4} - 80 x^{2} + 1024 = 0$ is $x = 8, - 8, 4, - 4$ .

$x = 8, - 8, 4, - 4$

Step 13

Use each root to create test intervals.

$x < - 8$

$- 8 < x < - 4$

$- 4 < x < 4$

$4 < x < 8$

$x > 8$

Step 14

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 14.1

Test a value on the interval $x < - 8$ to see if it makes the inequality true.

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Step 14.1.1

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Step 14.1.2

Replace $x$ with $- 10$ in the original inequality.

${(- 10)}^{4} - 80 {(- 10)}^{2} + 1024 < 0$

Step 14.1.3

The left side $3024$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 14.2

Test a value on the interval $- 8 < x < - 4$ to see if it makes the inequality true.

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Step 14.2.1

Choose a value on the interval $- 8 < x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 14.2.2

Replace $x$ with $- 6$ in the original inequality.

${(- 6)}^{4} - 80 {(- 6)}^{2} + 1024 < 0$

Step 14.2.3

The left side $- 560$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 14.3

Test a value on the interval $- 4 < x < 4$ to see if it makes the inequality true.

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Step 14.3.1

Choose a value on the interval $- 4 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 14.3.2

Replace $x$ with $0$ in the original inequality.

${(0)}^{4} - 80 {(0)}^{2} + 1024 < 0$

Step 14.3.3

The left side $1024$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 14.4

Test a value on the interval $4 < x < 8$ to see if it makes the inequality true.

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Step 14.4.1

Choose a value on the interval $4 < x < 8$ and see if this value makes the original inequality true.

$x = 6$

Step 14.4.2

Replace $x$ with $6$ in the original inequality.

${(6)}^{4} - 80 {(6)}^{2} + 1024 < 0$

Step 14.4.3

The left side $- 560$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 14.5

Test a value on the interval $x > 8$ to see if it makes the inequality true.

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Step 14.5.1

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Step 14.5.2

Replace $x$ with $10$ in the original inequality.

${(10)}^{4} - 80 {(10)}^{2} + 1024 < 0$

Step 14.5.3

The left side $3024$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 14.6

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 8$ False

$- 8 < x < - 4$ True

$- 4 < x < 4$ False

$4 < x < 8$ True

$x > 8$ False

$x < - 8$ False

$- 8 < x < - 4$ True

$- 4 < x < 4$ False

$4 < x < 8$ True

$x > 8$ False

Step 15

The solution consists of all of the true intervals.

$- 8 < x < - 4$ or $4 < x < 8$

Step 16

Convert the inequality to interval notation.

$(- 8, - 4) \cup (4, 8)$

Step 17