Trigonometry Examples

$x^{4} - 65 x^{2} + 784 < 0$

Step 1

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

$u^{2} - 65 u + 784 = 0$

$u = x^{2}$

Step 2

Factor $u^{2} - 65 u + 784$ using the AC method.

Tap for more steps...

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 $784$ and whose sum is $- 65$ .

$- 49, - 16$

Step 2.2

Write the factored form using these integers.

$(u - 49) (u - 16) = 0$

Step 3

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

$u - 49 = 0$

$u - 16 = 0$

Step 4

Set $u - 49$ equal to $0$ and solve for $u$ .

Tap for more steps...

Step 4.1

Set $u - 49$ equal to $0$ .

$u - 49 = 0$

Step 4.2

Add $49$ to both sides of the equation.

$u = 49$

Step 5

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

Tap for more steps...

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 $(u - 49) (u - 16) = 0$ true.

$u = 49, 16$

Step 7

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

$x^{2} = 49$

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

Step 8

Solve the first equation for $x$ .

$x^{2} = 49$

Step 9

Solve the equation for $x$ .

Tap for more steps...

Step 9.1

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

$x = \pm \sqrt{49}$

Step 9.2

Simplify $\pm \sqrt{49}$ .

Tap for more steps...

Step 9.2.1

Rewrite $49$ as $7^{2}$ .

$x = \pm \sqrt{7^{2}}$

Step 9.2.2

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

$x = \pm 7$

Step 9.3

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

Tap for more steps...

Step 9.3.1

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

$x = 7$

Step 9.3.2

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

$x = - 7$

Step 9.3.3

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

$x = 7, - 7$

Step 10

Solve the second equation for $x$ .

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

Step 11

Solve the equation for $x$ .

Tap for more steps...

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 $\pm \sqrt{16}$ .

Tap for more steps...

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.

Tap for more steps...

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 $x^{4} - 65 x^{2} + 784 = 0$ is $x = 7, - 7, 4, - 4$ .

$x = 7, - 7, 4, - 4$

Step 13

Use each root to create test intervals.

$x < - 7$

$- 7 < x < - 4$

$- 4 < x < 4$

$4 < x < 7$

$x > 7$

Step 14

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

Tap for more steps...

Step 14.1

Test a value on the interval $x < - 7$ to see if it makes the inequality true.

Tap for more steps...

Step 14.1.1

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Step 14.1.2

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

${(- 10)}^{4} - 65 {(- 10)}^{2} + 784 < 0$

Step 14.1.3

The left side $4284$ 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 $- 7 < x < - 4$ to see if it makes the inequality true.

Tap for more steps...

Step 14.2.1

Choose a value on the interval $- 7 < x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 14.2.2

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

${(- 6)}^{4} - 65 {(- 6)}^{2} + 784 < 0$

Step 14.2.3

The left side $- 260$ 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.

Tap for more steps...

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.

${(0)}^{4} - 65 {(0)}^{2} + 784 < 0$

Step 14.3.3

The left side $784$ 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 $4 < x < 7$ to see if it makes the inequality true.

Tap for more steps...

Step 14.4.1

Choose a value on the interval $4 < x < 7$ and see if this value makes the original inequality true.

$x = 6$

Step 14.4.2

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

${(6)}^{4} - 65 {(6)}^{2} + 784 < 0$

Step 14.4.3

The left side $- 260$ 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 $x > 7$ to see if it makes the inequality true.

Tap for more steps...

Step 14.5.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 14.5.2

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

${(10)}^{4} - 65 {(10)}^{2} + 784 < 0$

Step 14.5.3

The left side $4284$ 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.

$x < - 7$ False

$- 7 < x < - 4$ True

$- 4 < x < 4$ False

$4 < x < 7$ True

$x > 7$ False

$x < - 7$ False

$- 7 < x < - 4$ True

$- 4 < x < 4$ False

$4 < x < 7$ True

$x > 7$ False

Step 15

The solution consists of all of the true intervals.

$- 7 < x < - 4$ or $4 < x < 7$

Step 16

The result can be shown in multiple forms.

Inequality Form:

$- 7 < x < - 4 or 4 < x < 7$

Interval Notation:

$(- 7, - 4) \cup (4, 7)$

Step 17