Precalculus Examples

Solve for ? 2cos(x)^2-cos(x)-1=0
Step 1
Substitute for .
Step 2
Factor by grouping.
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Step 2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 2.1.1
Factor out of .
Step 2.1.2
Rewrite as plus
Step 2.1.3
Apply the distributive property.
Step 2.1.4
Multiply by .
Step 2.2
Factor out the greatest common factor from each group.
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Step 2.2.1
Group the first two terms and the last two terms.
Step 2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.3
Factor the polynomial by factoring out the greatest common factor, .
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
Solve for .
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Step 4.2.1
Subtract from both sides of the equation.
Step 4.2.2
Divide each term in by and simplify.
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Step 4.2.2.1
Divide each term in by .
Step 4.2.2.2
Simplify the left side.
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Step 4.2.2.2.1
Cancel the common factor of .
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Step 4.2.2.2.1.1
Cancel the common factor.
Step 4.2.2.2.1.2
Divide by .
Step 4.2.2.3
Simplify the right side.
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Step 4.2.2.3.1
Move the negative in front of the fraction.
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 for .
Step 8
Set up each of the solutions to solve for .
Step 9
Solve for in .
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Step 9.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 9.2
Simplify the right side.
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Step 9.2.1
The exact value of is .
Step 9.3
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 9.4
Simplify .
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Step 9.4.1
To write as a fraction with a common denominator, multiply by .
Step 9.4.2
Combine fractions.
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Step 9.4.2.1
Combine and .
Step 9.4.2.2
Combine the numerators over the common denominator.
Step 9.4.3
Simplify the numerator.
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Step 9.4.3.1
Multiply by .
Step 9.4.3.2
Subtract from .
Step 9.5
Find the period of .
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Step 9.5.1
The period of the function can be calculated using .
Step 9.5.2
Replace with in the formula for period.
Step 9.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9.5.4
Divide by .
Step 9.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 10
Solve for in .
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Step 10.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.2
Simplify the right side.
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Step 10.2.1
The exact value of is .
Step 10.3
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 10.4
Subtract from .
Step 10.5
Find the period of .
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Step 10.5.1
The period of the function can be calculated using .
Step 10.5.2
Replace with in the formula for period.
Step 10.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.5.4
Divide by .
Step 10.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 11
List all of the solutions.
, for any integer
Step 12
Consolidate the answers.
, for any integer