Finite Math Examples

Solve by Factoring 2x^(2/3)-11x^(1/3)-6=0
Step 1
Rewrite as .
Step 2
Let . Substitute for all occurrences of .
Step 3
Factor by grouping.
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Step 3.1
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 3.1.1
Factor out of .
Step 3.1.2
Rewrite as plus
Step 3.1.3
Apply the distributive property.
Step 3.1.4
Multiply by .
Step 3.2
Factor out the greatest common factor from each group.
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Step 3.2.1
Group the first two terms and the last two terms.
Step 3.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4
Replace all occurrences of with .
Step 5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 6
Set equal to and solve for .
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Step 6.1
Set equal to .
Step 6.2
Solve for .
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Step 6.2.1
Subtract from both sides of the equation.
Step 6.2.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 6.2.3
Simplify the exponent.
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Step 6.2.3.1
Simplify the left side.
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Step 6.2.3.1.1
Simplify .
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Step 6.2.3.1.1.1
Apply the product rule to .
Step 6.2.3.1.1.2
Raise to the power of .
Step 6.2.3.1.1.3
Multiply the exponents in .
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Step 6.2.3.1.1.3.1
Apply the power rule and multiply exponents, .
Step 6.2.3.1.1.3.2
Cancel the common factor of .
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Step 6.2.3.1.1.3.2.1
Cancel the common factor.
Step 6.2.3.1.1.3.2.2
Rewrite the expression.
Step 6.2.3.1.1.4
Simplify.
Step 6.2.3.2
Simplify the right side.
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Step 6.2.3.2.1
Raise to the power of .
Step 6.2.4
Divide each term in by and simplify.
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Step 6.2.4.1
Divide each term in by .
Step 6.2.4.2
Simplify the left side.
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Step 6.2.4.2.1
Cancel the common factor of .
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Step 6.2.4.2.1.1
Cancel the common factor.
Step 6.2.4.2.1.2
Divide by .
Step 6.2.4.3
Simplify the right side.
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Step 6.2.4.3.1
Move the negative in front of the fraction.
Step 7
Set equal to and solve for .
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Step 7.1
Set equal to .
Step 7.2
Solve for .
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Step 7.2.1
Add to both sides of the equation.
Step 7.2.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 7.2.3
Simplify the exponent.
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Step 7.2.3.1
Simplify the left side.
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Step 7.2.3.1.1
Simplify .
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Step 7.2.3.1.1.1
Multiply the exponents in .
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Step 7.2.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 7.2.3.1.1.1.2
Cancel the common factor of .
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Step 7.2.3.1.1.1.2.1
Cancel the common factor.
Step 7.2.3.1.1.1.2.2
Rewrite the expression.
Step 7.2.3.1.1.2
Simplify.
Step 7.2.3.2
Simplify the right side.
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Step 7.2.3.2.1
Raise to the power of .
Step 8
The final solution is all the values that make true.