Pre-Algebra Examples

Find the Quadratic Constant of Variation 36x^2+81y^2+504x-324y-828=0
Step 1
Use the quadratic formula to find the solutions.
Step 2
Substitute the values , , and into the quadratic formula and solve for .
Step 3
Simplify.
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Step 3.1
Simplify the numerator.
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Step 3.1.1
Raise to the power of .
Step 3.1.2
Multiply by .
Step 3.1.3
Apply the distributive property.
Step 3.1.4
Simplify.
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Step 3.1.4.1
Multiply by .
Step 3.1.4.2
Multiply by .
Step 3.1.4.3
Multiply by .
Step 3.1.5
Add and .
Step 3.1.6
Rewrite in a factored form.
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Step 3.1.6.1
Factor out of .
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Step 3.1.6.1.1
Factor out of .
Step 3.1.6.1.2
Factor out of .
Step 3.1.6.1.3
Factor out of .
Step 3.1.6.1.4
Factor out of .
Step 3.1.6.1.5
Factor out of .
Step 3.1.6.2
Factor by grouping.
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Step 3.1.6.2.1
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 3.1.6.2.1.1
Factor out of .
Step 3.1.6.2.1.2
Rewrite as plus
Step 3.1.6.2.1.3
Apply the distributive property.
Step 3.1.6.2.2
Factor out the greatest common factor from each group.
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Step 3.1.6.2.2.1
Group the first two terms and the last two terms.
Step 3.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3.1.7
Rewrite as .
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Step 3.1.7.1
Rewrite as .
Step 3.1.7.2
Rewrite as .
Step 3.1.7.3
Add parentheses.
Step 3.1.8
Pull terms out from under the radical.
Step 3.1.9
Raise to the power of .
Step 3.2
Multiply by .
Step 3.3
Simplify .
Step 4
Simplify the expression to solve for the portion of the .
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Step 4.1
Simplify the numerator.
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Step 4.1.1
Raise to the power of .
Step 4.1.2
Multiply by .
Step 4.1.3
Apply the distributive property.
Step 4.1.4
Simplify.
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Step 4.1.4.1
Multiply by .
Step 4.1.4.2
Multiply by .
Step 4.1.4.3
Multiply by .
Step 4.1.5
Add and .
Step 4.1.6
Rewrite in a factored form.
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Step 4.1.6.1
Factor out of .
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Step 4.1.6.1.1
Factor out of .
Step 4.1.6.1.2
Factor out of .
Step 4.1.6.1.3
Factor out of .
Step 4.1.6.1.4
Factor out of .
Step 4.1.6.1.5
Factor out of .
Step 4.1.6.2
Factor by grouping.
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Step 4.1.6.2.1
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 4.1.6.2.1.1
Factor out of .
Step 4.1.6.2.1.2
Rewrite as plus
Step 4.1.6.2.1.3
Apply the distributive property.
Step 4.1.6.2.2
Factor out the greatest common factor from each group.
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Step 4.1.6.2.2.1
Group the first two terms and the last two terms.
Step 4.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 4.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4.1.7
Rewrite as .
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Step 4.1.7.1
Rewrite as .
Step 4.1.7.2
Rewrite as .
Step 4.1.7.3
Add parentheses.
Step 4.1.8
Pull terms out from under the radical.
Step 4.1.9
Raise to the power of .
Step 4.2
Multiply by .
Step 4.3
Simplify .
Step 4.4
Change the to .
Step 4.5
Factor out of .
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Step 4.5.1
Factor out of .
Step 4.5.2
Factor out of .
Step 5
Simplify the expression to solve for the portion of the .
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Step 5.1
Simplify the numerator.
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Step 5.1.1
Raise to the power of .
Step 5.1.2
Multiply by .
Step 5.1.3
Apply the distributive property.
Step 5.1.4
Simplify.
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Step 5.1.4.1
Multiply by .
Step 5.1.4.2
Multiply by .
Step 5.1.4.3
Multiply by .
Step 5.1.5
Add and .
Step 5.1.6
Rewrite in a factored form.
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Step 5.1.6.1
Factor out of .
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Step 5.1.6.1.1
Factor out of .
Step 5.1.6.1.2
Factor out of .
Step 5.1.6.1.3
Factor out of .
Step 5.1.6.1.4
Factor out of .
Step 5.1.6.1.5
Factor out of .
Step 5.1.6.2
Factor by grouping.
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Step 5.1.6.2.1
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 5.1.6.2.1.1
Factor out of .
Step 5.1.6.2.1.2
Rewrite as plus
Step 5.1.6.2.1.3
Apply the distributive property.
Step 5.1.6.2.2
Factor out the greatest common factor from each group.
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Step 5.1.6.2.2.1
Group the first two terms and the last two terms.
Step 5.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.1.7
Rewrite as .
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Step 5.1.7.1
Rewrite as .
Step 5.1.7.2
Rewrite as .
Step 5.1.7.3
Add parentheses.
Step 5.1.8
Pull terms out from under the radical.
Step 5.1.9
Raise to the power of .
Step 5.2
Multiply by .
Step 5.3
Simplify .
Step 5.4
Change the to .
Step 5.5
Factor out of .
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Step 5.5.1
Factor out of .
Step 5.5.2
Factor out of .
Step 5.5.3
Factor out of .
Step 6
The final answer is the combination of both solutions.
Step 7
The given equation can not be written as , so doesn't vary directly with .
doesn't vary directly with