Trigonometry Examples

Solve for x x^4-65x^2+784<0
Step 1
Substitute into the equation. This will make the quadratic formula easy to use.
Step 2
Factor using the AC method.
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Step 2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.2
Write the factored form using these integers.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Add to both sides of the equation.
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Add to both sides of the equation.
Step 6
The final solution is all the values that make true.
Step 7
Substitute the real value of back into the solved equation.
Step 8
Solve the first equation for .
Step 9
Solve the equation for .
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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.
Step 9.2
Simplify .
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Step 9.2.1
Rewrite as .
Step 9.2.2
Pull terms out from under the radical, assuming positive real numbers.
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 to find the first solution.
Step 9.3.2
Next, use the negative value of the to find the second solution.
Step 9.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 10
Solve the second equation for .
Step 11
Solve the equation for .
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Step 11.1
Remove parentheses.
Step 11.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 11.3
Simplify .
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Step 11.3.1
Rewrite as .
Step 11.3.2
Pull terms out from under the radical, assuming positive real numbers.
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 to find the first solution.
Step 11.4.2
Next, use the negative value of the to find the second solution.
Step 11.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 12
The solution to is .
Step 13
Use each root to create test intervals.
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 to see if it makes the inequality true.
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Step 14.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 14.1.2
Replace with in the original inequality.
Step 14.1.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 14.2
Test a value on the interval to see if it makes the inequality true.
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Step 14.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 14.2.2
Replace with in the original inequality.
Step 14.2.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 14.3
Test a value on the interval to see if it makes the inequality true.
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Step 14.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 14.3.2
Replace with in the original inequality.
Step 14.3.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 14.4
Test a value on the interval to see if it makes the inequality true.
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Step 14.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 14.4.2
Replace with in the original inequality.
Step 14.4.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 14.5
Test a value on the interval to see if it makes the inequality true.
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Step 14.5.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 14.5.2
Replace with in the original inequality.
Step 14.5.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 14.6
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
True
False
False
True
False
True
False
Step 15
The solution consists of all of the true intervals.
or
Step 16
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 17