Finite Math Examples

Solve for x x^6+28x^3+27=0
Step 1
Factor the left side of the equation.
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Step 1.1
Rewrite as .
Step 1.2
Let . Substitute for all occurrences of .
Step 1.3
Factor using the AC method.
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Step 1.3.1
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 1.3.2
Write the factored form using these integers.
Step 1.4
Replace all occurrences of with .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Set equal to and solve for .
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Step 3.1
Set equal to .
Step 3.2
Solve for .
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Step 3.2.1
Subtract from both sides of the equation.
Step 3.2.2
Add to both sides of the equation.
Step 3.2.3
Factor the left side of the equation.
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Step 3.2.3.1
Rewrite as .
Step 3.2.3.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 3.2.3.3
Simplify.
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Step 3.2.3.3.1
Multiply by .
Step 3.2.3.3.2
One to any power is one.
Step 3.2.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.2.5
Set equal to and solve for .
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Step 3.2.5.1
Set equal to .
Step 3.2.5.2
Subtract from both sides of the equation.
Step 3.2.6
Set equal to and solve for .
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Step 3.2.6.1
Set equal to .
Step 3.2.6.2
Solve for .
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Step 3.2.6.2.1
Use the quadratic formula to find the solutions.
Step 3.2.6.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 3.2.6.2.3
Simplify.
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Step 3.2.6.2.3.1
Simplify the numerator.
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Step 3.2.6.2.3.1.1
Raise to the power of .
Step 3.2.6.2.3.1.2
Multiply .
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Step 3.2.6.2.3.1.2.1
Multiply by .
Step 3.2.6.2.3.1.2.2
Multiply by .
Step 3.2.6.2.3.1.3
Subtract from .
Step 3.2.6.2.3.1.4
Rewrite as .
Step 3.2.6.2.3.1.5
Rewrite as .
Step 3.2.6.2.3.1.6
Rewrite as .
Step 3.2.6.2.3.2
Multiply by .
Step 3.2.6.2.4
The final answer is the combination of both solutions.
Step 3.2.7
The final solution is all the values that make true.
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
Add to both sides of the equation.
Step 4.2.3
Factor the left side of the equation.
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Step 4.2.3.1
Rewrite as .
Step 4.2.3.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 4.2.3.3
Simplify.
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Step 4.2.3.3.1
Multiply by .
Step 4.2.3.3.2
Raise to the power of .
Step 4.2.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.2.5
Set equal to and solve for .
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Step 4.2.5.1
Set equal to .
Step 4.2.5.2
Subtract from both sides of the equation.
Step 4.2.6
Set equal to and solve for .
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Step 4.2.6.1
Set equal to .
Step 4.2.6.2
Solve for .
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Step 4.2.6.2.1
Use the quadratic formula to find the solutions.
Step 4.2.6.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 4.2.6.2.3
Simplify.
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Step 4.2.6.2.3.1
Simplify the numerator.
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Step 4.2.6.2.3.1.1
Raise to the power of .
Step 4.2.6.2.3.1.2
Multiply .
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Step 4.2.6.2.3.1.2.1
Multiply by .
Step 4.2.6.2.3.1.2.2
Multiply by .
Step 4.2.6.2.3.1.3
Subtract from .
Step 4.2.6.2.3.1.4
Rewrite as .
Step 4.2.6.2.3.1.5
Rewrite as .
Step 4.2.6.2.3.1.6
Rewrite as .
Step 4.2.6.2.3.1.7
Rewrite as .
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Step 4.2.6.2.3.1.7.1
Factor out of .
Step 4.2.6.2.3.1.7.2
Rewrite as .
Step 4.2.6.2.3.1.8
Pull terms out from under the radical.
Step 4.2.6.2.3.1.9
Move to the left of .
Step 4.2.6.2.3.2
Multiply by .
Step 4.2.6.2.4
The final answer is the combination of both solutions.
Step 4.2.7
The final solution is all the values that make true.
Step 5
The final solution is all the values that make true.