Trigonometry Examples

Find the Roots (Zeros) p(x)=27x^3-64
Step 1
Set equal to .
Step 2
Solve for .
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Step 2.1
Add to both sides of the equation.
Step 2.2
Subtract from both sides of the equation.
Step 2.3
Factor the left side of the equation.
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Step 2.3.1
Rewrite as .
Step 2.3.2
Rewrite as .
Step 2.3.3
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 2.3.4
Simplify.
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Step 2.3.4.1
Apply the product rule to .
Step 2.3.4.2
Raise to the power of .
Step 2.3.4.3
Multiply by .
Step 2.3.4.4
Raise to the power of .
Step 2.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.5
Set equal to and solve for .
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Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
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Step 2.5.2.1
Add to both sides of the equation.
Step 2.5.2.2
Divide each term in by and simplify.
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Step 2.5.2.2.1
Divide each term in by .
Step 2.5.2.2.2
Simplify the left side.
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Step 2.5.2.2.2.1
Cancel the common factor of .
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Step 2.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.2.1.2
Divide by .
Step 2.6
Set equal to and solve for .
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Step 2.6.1
Set equal to .
Step 2.6.2
Solve for .
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Step 2.6.2.1
Use the quadratic formula to find the solutions.
Step 2.6.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 2.6.2.3
Simplify.
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Step 2.6.2.3.1
Simplify the numerator.
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Step 2.6.2.3.1.1
Raise to the power of .
Step 2.6.2.3.1.2
Multiply .
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Step 2.6.2.3.1.2.1
Multiply by .
Step 2.6.2.3.1.2.2
Multiply by .
Step 2.6.2.3.1.3
Subtract from .
Step 2.6.2.3.1.4
Rewrite as .
Step 2.6.2.3.1.5
Rewrite as .
Step 2.6.2.3.1.6
Rewrite as .
Step 2.6.2.3.1.7
Rewrite as .
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Step 2.6.2.3.1.7.1
Factor out of .
Step 2.6.2.3.1.7.2
Rewrite as .
Step 2.6.2.3.1.8
Pull terms out from under the radical.
Step 2.6.2.3.1.9
Move to the left of .
Step 2.6.2.3.2
Multiply by .
Step 2.6.2.3.3
Simplify .
Step 2.6.2.4
The final answer is the combination of both solutions.
Step 2.7
The final solution is all the values that make true.
Step 3