Calculus Examples

Solve for x 1/(x^4+4)+1/(x^4-4x^3+4x^2-4)-2/(x^4-4x^2-8x-4)=0
Step 1
Factor each term.
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Step 1.1
Factor using the perfect square rule.
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Step 1.1.1
Rewrite as .
Step 1.1.2
Rewrite as .
Step 1.1.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 1.1.4
Rewrite the polynomial.
Step 1.1.5
Factor using the perfect square trinomial rule , where and .
Step 1.2
Rewrite as .
Step 1.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2
Find the LCD of the terms in the equation.
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Step 2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.2
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 2.3
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.4
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.5
The factor for is itself.
occurs time.
Step 2.6
The factor for is itself.
occurs time.
Step 2.7
The factor for is itself.
occurs time.
Step 2.8
The factor for is itself.
occurs time.
Step 2.9
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Step 3
Multiply each term in by to eliminate the fractions.
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Step 3.1
Multiply each term in by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Cancel the common factor of .
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Step 3.2.1.1.1
Factor out of .
Step 3.2.1.1.2
Cancel the common factor.
Step 3.2.1.1.3
Rewrite the expression.
Step 3.2.1.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.2.1.3
Combine the opposite terms in .
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Step 3.2.1.3.1
Reorder the factors in the terms and .
Step 3.2.1.3.2
Add and .
Step 3.2.1.3.3
Add and .
Step 3.2.1.4
Simplify each term.
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Step 3.2.1.4.1
Multiply by by adding the exponents.
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Step 3.2.1.4.1.1
Use the power rule to combine exponents.
Step 3.2.1.4.1.2
Add and .
Step 3.2.1.4.2
Rewrite using the commutative property of multiplication.
Step 3.2.1.4.3
Multiply by by adding the exponents.
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Step 3.2.1.4.3.1
Move .
Step 3.2.1.4.3.2
Multiply by .
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Step 3.2.1.4.3.2.1
Raise to the power of .
Step 3.2.1.4.3.2.2
Use the power rule to combine exponents.
Step 3.2.1.4.3.3
Add and .
Step 3.2.1.4.4
Multiply by by adding the exponents.
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Step 3.2.1.4.4.1
Move .
Step 3.2.1.4.4.2
Multiply by .
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Step 3.2.1.4.4.2.1
Raise to the power of .
Step 3.2.1.4.4.2.2
Use the power rule to combine exponents.
Step 3.2.1.4.4.3
Add and .
Step 3.2.1.4.5
Rewrite using the commutative property of multiplication.
Step 3.2.1.4.6
Multiply by by adding the exponents.
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Step 3.2.1.4.6.1
Move .
Step 3.2.1.4.6.2
Multiply by .
Step 3.2.1.4.7
Multiply by .
Step 3.2.1.4.8
Multiply by .
Step 3.2.1.4.9
Multiply by .
Step 3.2.1.4.10
Multiply by .
Step 3.2.1.5
Combine the opposite terms in .
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Step 3.2.1.5.1
Subtract from .
Step 3.2.1.5.2
Add and .
Step 3.2.1.6
Subtract from .
Step 3.2.1.7
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.2.1.8
Combine the opposite terms in .
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Step 3.2.1.8.1
Reorder the factors in the terms and .
Step 3.2.1.8.2
Add and .
Step 3.2.1.8.3
Add and .
Step 3.2.1.9
Simplify each term.
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Step 3.2.1.9.1
Multiply by by adding the exponents.
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Step 3.2.1.9.1.1
Use the power rule to combine exponents.
Step 3.2.1.9.1.2
Add and .
Step 3.2.1.9.2
Rewrite using the commutative property of multiplication.
Step 3.2.1.9.3
Multiply by by adding the exponents.
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Step 3.2.1.9.3.1
Move .
Step 3.2.1.9.3.2
Multiply by .
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Step 3.2.1.9.3.2.1
Raise to the power of .
Step 3.2.1.9.3.2.2
Use the power rule to combine exponents.
Step 3.2.1.9.3.3
Add and .
Step 3.2.1.9.4
Move to the left of .
Step 3.2.1.9.5
Multiply by by adding the exponents.
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Step 3.2.1.9.5.1
Move .
Step 3.2.1.9.5.2
Use the power rule to combine exponents.
Step 3.2.1.9.5.3
Add and .
Step 3.2.1.9.6
Rewrite using the commutative property of multiplication.
Step 3.2.1.9.7
Multiply by by adding the exponents.
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Step 3.2.1.9.7.1
Move .
Step 3.2.1.9.7.2
Use the power rule to combine exponents.
Step 3.2.1.9.7.3
Add and .
Step 3.2.1.9.8
Multiply by .
Step 3.2.1.9.9
Rewrite using the commutative property of multiplication.
Step 3.2.1.9.10
Multiply by by adding the exponents.
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Step 3.2.1.9.10.1
Move .
Step 3.2.1.9.10.2
Multiply by .
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Step 3.2.1.9.10.2.1
Raise to the power of .
Step 3.2.1.9.10.2.2
Use the power rule to combine exponents.
Step 3.2.1.9.10.3
Add and .
Step 3.2.1.9.11
Multiply by .
Step 3.2.1.9.12
Multiply by .
Step 3.2.1.9.13
Rewrite using the commutative property of multiplication.
Step 3.2.1.9.14
Multiply by by adding the exponents.
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Step 3.2.1.9.14.1
Move .
Step 3.2.1.9.14.2
Use the power rule to combine exponents.
Step 3.2.1.9.14.3
Add and .
Step 3.2.1.9.15
Multiply by .
Step 3.2.1.9.16
Rewrite using the commutative property of multiplication.
Step 3.2.1.9.17
Multiply by by adding the exponents.
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Step 3.2.1.9.17.1
Move .
Step 3.2.1.9.17.2
Multiply by .
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Step 3.2.1.9.17.2.1
Raise to the power of .
Step 3.2.1.9.17.2.2
Use the power rule to combine exponents.
Step 3.2.1.9.17.3
Add and .
Step 3.2.1.9.18
Multiply by .
Step 3.2.1.9.19
Multiply by .
Step 3.2.1.9.20
Multiply by .
Step 3.2.1.9.21
Multiply by .
Step 3.2.1.9.22
Multiply by .
Step 3.2.1.10
Combine the opposite terms in .
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Step 3.2.1.10.1
Add and .
Step 3.2.1.10.2
Add and .
Step 3.2.1.11
Add and .
Step 3.2.1.12
Add and .
Step 3.2.1.13
Subtract from .
Step 3.2.1.14
Subtract from .
Step 3.2.1.15
Subtract from .
Step 3.2.1.16
Cancel the common factor of .
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Step 3.2.1.16.1
Factor out of .
Step 3.2.1.16.2
Cancel the common factor.
Step 3.2.1.16.3
Rewrite the expression.
Step 3.2.1.17
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.2.1.18
Combine the opposite terms in .
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Step 3.2.1.18.1
Reorder the factors in the terms and .
Step 3.2.1.18.2
Add and .
Step 3.2.1.18.3
Add and .
Step 3.2.1.19
Simplify each term.
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Step 3.2.1.19.1
Multiply by by adding the exponents.
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Step 3.2.1.19.1.1
Use the power rule to combine exponents.
Step 3.2.1.19.1.2
Add and .
Step 3.2.1.19.2
Rewrite using the commutative property of multiplication.
Step 3.2.1.19.3
Multiply by by adding the exponents.
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Step 3.2.1.19.3.1
Move .
Step 3.2.1.19.3.2
Use the power rule to combine exponents.
Step 3.2.1.19.3.3
Add and .
Step 3.2.1.19.4
Rewrite using the commutative property of multiplication.
Step 3.2.1.19.5
Multiply by by adding the exponents.
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Step 3.2.1.19.5.1
Move .
Step 3.2.1.19.5.2
Multiply by .
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Step 3.2.1.19.5.2.1
Raise to the power of .
Step 3.2.1.19.5.2.2
Use the power rule to combine exponents.
Step 3.2.1.19.5.3
Add and .
Step 3.2.1.19.6
Multiply by .
Step 3.2.1.19.7
Multiply by .
Step 3.2.1.19.8
Multiply by .
Step 3.2.1.20
Cancel the common factor of .
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Step 3.2.1.20.1
Move the leading negative in into the numerator.
Step 3.2.1.20.2
Factor out of .
Step 3.2.1.20.3
Cancel the common factor.
Step 3.2.1.20.4
Rewrite the expression.
Step 3.2.1.21
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.2.1.22
Simplify each term.
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Step 3.2.1.22.1
Multiply by by adding the exponents.
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Step 3.2.1.22.1.1
Use the power rule to combine exponents.
Step 3.2.1.22.1.2
Add and .
Step 3.2.1.22.2
Rewrite using the commutative property of multiplication.
Step 3.2.1.22.3
Multiply by by adding the exponents.
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Step 3.2.1.22.3.1
Move .
Step 3.2.1.22.3.2
Multiply by .
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Step 3.2.1.22.3.2.1
Raise to the power of .
Step 3.2.1.22.3.2.2
Use the power rule to combine exponents.
Step 3.2.1.22.3.3
Add and .
Step 3.2.1.22.4
Move to the left of .
Step 3.2.1.22.5
Multiply by .
Step 3.2.1.22.6
Multiply by .
Step 3.2.1.23
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.2.1.24
Simplify each term.
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Step 3.2.1.24.1
Multiply by by adding the exponents.
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Step 3.2.1.24.1.1
Use the power rule to combine exponents.
Step 3.2.1.24.1.2
Add and .
Step 3.2.1.24.2
Rewrite using the commutative property of multiplication.
Step 3.2.1.24.3
Multiply by by adding the exponents.
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Step 3.2.1.24.3.1
Move .
Step 3.2.1.24.3.2
Multiply by .
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Step 3.2.1.24.3.2.1
Raise to the power of .
Step 3.2.1.24.3.2.2
Use the power rule to combine exponents.
Step 3.2.1.24.3.3
Add and .
Step 3.2.1.24.4
Move to the left of .
Step 3.2.1.24.5
Multiply by by adding the exponents.
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Step 3.2.1.24.5.1
Move .
Step 3.2.1.24.5.2
Use the power rule to combine exponents.
Step 3.2.1.24.5.3
Add and .
Step 3.2.1.24.6
Rewrite using the commutative property of multiplication.
Step 3.2.1.24.7
Multiply by by adding the exponents.
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Step 3.2.1.24.7.1
Move .
Step 3.2.1.24.7.2
Multiply by .
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Step 3.2.1.24.7.2.1
Raise to the power of .
Step 3.2.1.24.7.2.2
Use the power rule to combine exponents.
Step 3.2.1.24.7.3
Add and .
Step 3.2.1.24.8
Multiply by .
Step 3.2.1.24.9
Multiply by .
Step 3.2.1.24.10
Multiply by by adding the exponents.
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Step 3.2.1.24.10.1
Move .
Step 3.2.1.24.10.2
Use the power rule to combine exponents.
Step 3.2.1.24.10.3
Add and .
Step 3.2.1.24.11
Rewrite using the commutative property of multiplication.
Step 3.2.1.24.12
Multiply by by adding the exponents.
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Step 3.2.1.24.12.1
Move .
Step 3.2.1.24.12.2
Multiply by .
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Step 3.2.1.24.12.2.1
Raise to the power of .
Step 3.2.1.24.12.2.2
Use the power rule to combine exponents.
Step 3.2.1.24.12.3
Add and .
Step 3.2.1.24.13
Multiply by .
Step 3.2.1.24.14
Multiply by .
Step 3.2.1.24.15
Multiply by by adding the exponents.
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Step 3.2.1.24.15.1
Move .
Step 3.2.1.24.15.2
Use the power rule to combine exponents.
Step 3.2.1.24.15.3
Add and .
Step 3.2.1.24.16
Rewrite using the commutative property of multiplication.
Step 3.2.1.24.17
Multiply by by adding the exponents.
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Step 3.2.1.24.17.1
Move .
Step 3.2.1.24.17.2
Multiply by .
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Step 3.2.1.24.17.2.1
Raise to the power of .
Step 3.2.1.24.17.2.2
Use the power rule to combine exponents.
Step 3.2.1.24.17.3
Add and .
Step 3.2.1.24.18
Multiply by .
Step 3.2.1.24.19
Multiply by .
Step 3.2.1.24.20
Multiply by by adding the exponents.
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Step 3.2.1.24.20.1
Move .
Step 3.2.1.24.20.2
Multiply by .
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Step 3.2.1.24.20.2.1
Raise to the power of .
Step 3.2.1.24.20.2.2
Use the power rule to combine exponents.
Step 3.2.1.24.20.3
Add and .
Step 3.2.1.24.21
Rewrite using the commutative property of multiplication.
Step 3.2.1.24.22
Multiply by by adding the exponents.
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Step 3.2.1.24.22.1
Move .
Step 3.2.1.24.22.2
Multiply by .
Step 3.2.1.24.23
Multiply by .
Step 3.2.1.24.24
Multiply by .
Step 3.2.1.24.25
Multiply by .
Step 3.2.1.24.26
Multiply by .
Step 3.2.1.25
Combine the opposite terms in .
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Step 3.2.1.25.1
Add and .
Step 3.2.1.25.2
Add and .
Step 3.2.1.25.3
Subtract from .
Step 3.2.1.25.4
Add and .
Step 3.2.1.25.5
Add and .
Step 3.2.1.25.6
Add and .
Step 3.2.1.25.7
Add and .
Step 3.2.1.25.8
Add and .
Step 3.2.1.25.9
Subtract from .
Step 3.2.1.25.10
Add and .
Step 3.2.1.26
Subtract from .
Step 3.2.1.27
Subtract from .
Step 3.2.1.28
Apply the distributive property.
Step 3.2.1.29
Simplify.
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Step 3.2.1.29.1
Multiply by .
Step 3.2.1.29.2
Multiply by .
Step 3.2.1.29.3
Multiply by .
Step 3.2.1.29.4
Multiply by .
Step 3.2.1.29.5
Multiply by .
Step 3.2.2
Simplify by adding terms.
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Step 3.2.2.1
Combine the opposite terms in .
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Step 3.2.2.1.1
Subtract from .
Step 3.2.2.1.2
Add and .
Step 3.2.2.1.3
Subtract from .
Step 3.2.2.1.4
Add and .
Step 3.2.2.1.5
Subtract from .
Step 3.2.2.1.6
Add and .
Step 3.2.2.2
Add and .
Step 3.2.2.3
Combine the opposite terms in .
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Step 3.2.2.3.1
Subtract from .
Step 3.2.2.3.2
Add and .
Step 3.2.2.4
Add and .
Step 3.2.2.5
Add and .
Step 3.2.2.6
Subtract from .
Step 3.2.2.7
Subtract from .
Step 3.3
Simplify the right side.
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Step 3.3.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.3.2
Simplify each term.
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Step 3.3.2.1
Multiply by by adding the exponents.
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Step 3.3.2.1.1
Use the power rule to combine exponents.
Step 3.3.2.1.2
Add and .
Step 3.3.2.2
Rewrite using the commutative property of multiplication.
Step 3.3.2.3
Multiply by by adding the exponents.
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Step 3.3.2.3.1
Move .
Step 3.3.2.3.2
Multiply by .
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Step 3.3.2.3.2.1
Raise to the power of .
Step 3.3.2.3.2.2
Use the power rule to combine exponents.
Step 3.3.2.3.3
Add and .
Step 3.3.2.4
Move to the left of .
Step 3.3.2.5
Multiply by .
Step 3.3.2.6
Multiply by .
Step 3.3.3
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.3.4
Simplify terms.
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Step 3.3.4.1
Simplify each term.
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Step 3.3.4.1.1
Multiply by by adding the exponents.
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Step 3.3.4.1.1.1
Use the power rule to combine exponents.
Step 3.3.4.1.1.2
Add and .
Step 3.3.4.1.2
Rewrite using the commutative property of multiplication.
Step 3.3.4.1.3
Multiply by by adding the exponents.
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Step 3.3.4.1.3.1
Move .
Step 3.3.4.1.3.2
Multiply by .
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Step 3.3.4.1.3.2.1
Raise to the power of .
Step 3.3.4.1.3.2.2
Use the power rule to combine exponents.
Step 3.3.4.1.3.3
Add and .
Step 3.3.4.1.4
Move to the left of .
Step 3.3.4.1.5
Multiply by by adding the exponents.
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Step 3.3.4.1.5.1
Move .
Step 3.3.4.1.5.2
Use the power rule to combine exponents.
Step 3.3.4.1.5.3
Add and .
Step 3.3.4.1.6
Rewrite using the commutative property of multiplication.
Step 3.3.4.1.7
Multiply by by adding the exponents.
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Step 3.3.4.1.7.1
Move .
Step 3.3.4.1.7.2
Multiply by .
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Step 3.3.4.1.7.2.1
Raise to the power of .
Step 3.3.4.1.7.2.2
Use the power rule to combine exponents.
Step 3.3.4.1.7.3
Add and .
Step 3.3.4.1.8
Multiply by .
Step 3.3.4.1.9
Multiply by .
Step 3.3.4.1.10
Multiply by by adding the exponents.
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Step 3.3.4.1.10.1
Move .
Step 3.3.4.1.10.2
Use the power rule to combine exponents.
Step 3.3.4.1.10.3
Add and .
Step 3.3.4.1.11
Rewrite using the commutative property of multiplication.
Step 3.3.4.1.12
Multiply by by adding the exponents.
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Step 3.3.4.1.12.1
Move .
Step 3.3.4.1.12.2
Multiply by .
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Step 3.3.4.1.12.2.1
Raise to the power of .
Step 3.3.4.1.12.2.2
Use the power rule to combine exponents.
Step 3.3.4.1.12.3
Add and .
Step 3.3.4.1.13
Multiply by .
Step 3.3.4.1.14
Multiply by .
Step 3.3.4.1.15
Multiply by by adding the exponents.
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Step 3.3.4.1.15.1
Move .
Step 3.3.4.1.15.2
Use the power rule to combine exponents.
Step 3.3.4.1.15.3
Add and .
Step 3.3.4.1.16
Rewrite using the commutative property of multiplication.
Step 3.3.4.1.17
Multiply by by adding the exponents.
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Step 3.3.4.1.17.1
Move .
Step 3.3.4.1.17.2
Multiply by .
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Step 3.3.4.1.17.2.1
Raise to the power of .
Step 3.3.4.1.17.2.2
Use the power rule to combine exponents.
Step 3.3.4.1.17.3
Add and .
Step 3.3.4.1.18
Multiply by .
Step 3.3.4.1.19
Multiply by .
Step 3.3.4.1.20
Multiply by by adding the exponents.
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Step 3.3.4.1.20.1
Move .
Step 3.3.4.1.20.2
Multiply by .
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Step 3.3.4.1.20.2.1
Raise to the power of .
Step 3.3.4.1.20.2.2
Use the power rule to combine exponents.
Step 3.3.4.1.20.3
Add and .
Step 3.3.4.1.21
Rewrite using the commutative property of multiplication.
Step 3.3.4.1.22
Multiply by by adding the exponents.
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Step 3.3.4.1.22.1
Move .
Step 3.3.4.1.22.2
Multiply by .
Step 3.3.4.1.23
Multiply by .
Step 3.3.4.1.24
Multiply by .
Step 3.3.4.1.25
Multiply by .
Step 3.3.4.1.26
Multiply by .
Step 3.3.4.2
Simplify by adding terms.
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Step 3.3.4.2.1
Combine the opposite terms in .
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Step 3.3.4.2.1.1
Add and .
Step 3.3.4.2.1.2
Add and .
Step 3.3.4.2.1.3
Subtract from .
Step 3.3.4.2.1.4
Add and .
Step 3.3.4.2.1.5
Add and .
Step 3.3.4.2.1.6
Add and .
Step 3.3.4.2.1.7
Add and .
Step 3.3.4.2.1.8
Add and .
Step 3.3.4.2.1.9
Subtract from .
Step 3.3.4.2.1.10
Add and .
Step 3.3.4.2.2
Subtract from .
Step 3.3.4.2.3
Subtract from .
Step 3.3.5
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.3.6
Simplify terms.
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Step 3.3.6.1
Simplify each term.
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Step 3.3.6.1.1
Multiply by by adding the exponents.
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Step 3.3.6.1.1.1
Use the power rule to combine exponents.
Step 3.3.6.1.1.2
Add and .
Step 3.3.6.1.2
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.3
Multiply by by adding the exponents.
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Step 3.3.6.1.3.1
Move .
Step 3.3.6.1.3.2
Use the power rule to combine exponents.
Step 3.3.6.1.3.3
Add and .
Step 3.3.6.1.4
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.5
Multiply by by adding the exponents.
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Step 3.3.6.1.5.1
Move .
Step 3.3.6.1.5.2
Multiply by .
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Step 3.3.6.1.5.2.1
Raise to the power of .
Step 3.3.6.1.5.2.2
Use the power rule to combine exponents.
Step 3.3.6.1.5.3
Add and .
Step 3.3.6.1.6
Move to the left of .
Step 3.3.6.1.7
Multiply by by adding the exponents.
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Step 3.3.6.1.7.1
Move .
Step 3.3.6.1.7.2
Use the power rule to combine exponents.
Step 3.3.6.1.7.3
Add and .
Step 3.3.6.1.8
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.9
Multiply by by adding the exponents.
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Step 3.3.6.1.9.1
Move .
Step 3.3.6.1.9.2
Use the power rule to combine exponents.
Step 3.3.6.1.9.3
Add and .
Step 3.3.6.1.10
Multiply by .
Step 3.3.6.1.11
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.12
Multiply by by adding the exponents.
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Step 3.3.6.1.12.1
Move .
Step 3.3.6.1.12.2
Multiply by .
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Step 3.3.6.1.12.2.1
Raise to the power of .
Step 3.3.6.1.12.2.2
Use the power rule to combine exponents.
Step 3.3.6.1.12.3
Add and .
Step 3.3.6.1.13
Multiply by .
Step 3.3.6.1.14
Multiply by .
Step 3.3.6.1.15
Multiply by by adding the exponents.
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Step 3.3.6.1.15.1
Move .
Step 3.3.6.1.15.2
Use the power rule to combine exponents.
Step 3.3.6.1.15.3
Add and .
Step 3.3.6.1.16
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.17
Multiply by by adding the exponents.
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Step 3.3.6.1.17.1
Move .
Step 3.3.6.1.17.2
Use the power rule to combine exponents.
Step 3.3.6.1.17.3
Add and .
Step 3.3.6.1.18
Multiply by .
Step 3.3.6.1.19
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.20
Multiply by by adding the exponents.
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Step 3.3.6.1.20.1
Move .
Step 3.3.6.1.20.2
Multiply by .
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Step 3.3.6.1.20.2.1
Raise to the power of .
Step 3.3.6.1.20.2.2
Use the power rule to combine exponents.
Step 3.3.6.1.20.3
Add and .
Step 3.3.6.1.21
Multiply by .
Step 3.3.6.1.22
Multiply by .
Step 3.3.6.1.23
Multiply by by adding the exponents.
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Step 3.3.6.1.23.1
Move .
Step 3.3.6.1.23.2
Use the power rule to combine exponents.
Step 3.3.6.1.23.3
Add and .
Step 3.3.6.1.24
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.25
Multiply by by adding the exponents.
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Step 3.3.6.1.25.1
Move .
Step 3.3.6.1.25.2
Use the power rule to combine exponents.
Step 3.3.6.1.25.3
Add and .
Step 3.3.6.1.26
Multiply by .
Step 3.3.6.1.27
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.28
Multiply by by adding the exponents.
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Step 3.3.6.1.28.1
Move .
Step 3.3.6.1.28.2
Multiply by .
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Step 3.3.6.1.28.2.1
Raise to the power of .
Step 3.3.6.1.28.2.2
Use the power rule to combine exponents.
Step 3.3.6.1.28.3
Add and .
Step 3.3.6.1.29
Multiply by .
Step 3.3.6.1.30
Multiply by .
Step 3.3.6.1.31
Multiply by by adding the exponents.
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Step 3.3.6.1.31.1
Move .
Step 3.3.6.1.31.2
Use the power rule to combine exponents.
Step 3.3.6.1.31.3
Add and .
Step 3.3.6.1.32
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.33
Multiply by by adding the exponents.
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Step 3.3.6.1.33.1
Move .
Step 3.3.6.1.33.2
Use the power rule to combine exponents.
Step 3.3.6.1.33.3
Add and .
Step 3.3.6.1.34
Multiply by .
Step 3.3.6.1.35
Rewrite using the commutative property of multiplication.
Step 3.3.6.1.36
Multiply by by adding the exponents.
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Step 3.3.6.1.36.1
Move .
Step 3.3.6.1.36.2
Multiply by .
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Step 3.3.6.1.36.2.1
Raise to the power of .
Step 3.3.6.1.36.2.2
Use the power rule to combine exponents.
Step 3.3.6.1.36.3
Add and .
Step 3.3.6.1.37
Multiply by .
Step 3.3.6.1.38
Multiply by .
Step 3.3.6.1.39
Multiply by .
Step 3.3.6.1.40
Multiply by .
Step 3.3.6.1.41
Multiply by .
Step 3.3.6.2
Simplify by adding terms.
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Step 3.3.6.2.1
Combine the opposite terms in .
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Step 3.3.6.2.1.1
Add and .
Step 3.3.6.2.1.2
Add and .
Step 3.3.6.2.1.3
Subtract from .
Step 3.3.6.2.1.4
Add and .
Step 3.3.6.2.1.5
Add and .
Step 3.3.6.2.1.6
Add and .
Step 3.3.6.2.1.7
Add and .
Step 3.3.6.2.1.8
Add and .
Step 3.3.6.2.2
Add and .
Step 3.3.6.2.3
Add and .
Step 3.3.6.2.4
Subtract from .
Step 3.3.6.2.5
Subtract from .
Step 3.3.6.2.6
Subtract from .
Step 3.3.6.2.7
Subtract from .
Step 3.3.6.2.8
Multiply by .
Step 4
Solve the equation.
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Step 4.1
Factor the left side of the equation.
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Step 4.1.1
Factor out of .
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Step 4.1.1.1
Factor out of .
Step 4.1.1.2
Factor out of .
Step 4.1.1.3
Factor out of .
Step 4.1.1.4
Factor out of .
Step 4.1.1.5
Factor out of .
Step 4.1.1.6
Factor out of .
Step 4.1.1.7
Factor out of .
Step 4.1.1.8
Factor out of .
Step 4.1.1.9
Factor out of .
Step 4.1.1.10
Factor out of .
Step 4.1.1.11
Factor out of .
Step 4.1.2
Factor out of .
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Step 4.1.2.1
Factor out of .
Step 4.1.2.2
Factor out of .
Step 4.1.2.3
Factor out of .
Step 4.1.2.4
Factor out of .
Step 4.1.2.5
Factor out of .
Step 4.1.3
Reorder terms.
Step 4.1.4
Factor.
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Step 4.1.4.1
Factor using the rational roots test.
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Step 4.1.4.1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 4.1.4.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 4.1.4.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 4.1.4.1.3.1
Substitute into the polynomial.
Step 4.1.4.1.3.2
Raise to the power of .
Step 4.1.4.1.3.3
Raise to the power of .
Step 4.1.4.1.3.4
Multiply by .
Step 4.1.4.1.3.5
Subtract from .
Step 4.1.4.1.3.6
Add and .
Step 4.1.4.1.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 4.1.4.1.5
Divide by .
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Step 4.1.4.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
--++
Step 4.1.4.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
--++
Step 4.1.4.1.5.3
Multiply the new quotient term by the divisor.
--++
+-
Step 4.1.4.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
--++
-+
Step 4.1.4.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--++
-+
-
Step 4.1.4.1.5.6
Pull the next terms from the original dividend down into the current dividend.
--++
-+
-+
Step 4.1.4.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
-
--++
-+
-+
Step 4.1.4.1.5.8
Multiply the new quotient term by the divisor.
-
--++
-+
-+
-+
Step 4.1.4.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
-
--++
-+
-+
+-
Step 4.1.4.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
--++
-+
-+
+-
-
Step 4.1.4.1.5.11
Pull the next terms from the original dividend down into the current dividend.
-
--++
-+
-+
+-
-+
Step 4.1.4.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
--
--++
-+
-+
+-
-+
Step 4.1.4.1.5.13
Multiply the new quotient term by the divisor.
--
--++
-+
-+
+-
-+
-+
Step 4.1.4.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
--
--++
-+
-+
+-
-+
+-
Step 4.1.4.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--
--++
-+
-+
+-
-+
+-
Step 4.1.4.1.5.16
Since the remander is , the final answer is the quotient.
Step 4.1.4.1.6
Write as a set of factors.
Step 4.1.4.2
Remove unnecessary parentheses.
Step 4.1.5
Factor out of .
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Step 4.1.5.1
Factor out of .
Step 4.1.5.2
Factor out of .
Step 4.1.5.3
Factor out of .
Step 4.1.5.4
Factor out of .
Step 4.1.5.5
Factor out of .
Step 4.1.6
Factor.
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Step 4.1.6.1
Factor using the rational roots test.
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Step 4.1.6.1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 4.1.6.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 4.1.6.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 4.1.6.1.3.1
Substitute into the polynomial.
Step 4.1.6.1.3.2
Raise to the power of .
Step 4.1.6.1.3.3
Raise to the power of .
Step 4.1.6.1.3.4
Multiply by .
Step 4.1.6.1.3.5
Subtract from .
Step 4.1.6.1.3.6
Add and .
Step 4.1.6.1.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 4.1.6.1.5
Divide by .
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Step 4.1.6.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
--++
Step 4.1.6.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
--++
Step 4.1.6.1.5.3
Multiply the new quotient term by the divisor.
--++
+-
Step 4.1.6.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
--++
-+
Step 4.1.6.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--++
-+
-
Step 4.1.6.1.5.6
Pull the next terms from the original dividend down into the current dividend.
--++
-+
-+
Step 4.1.6.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
-
--++
-+
-+
Step 4.1.6.1.5.8
Multiply the new quotient term by the divisor.
-
--++
-+
-+
-+
Step 4.1.6.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
-
--++
-+
-+
+-
Step 4.1.6.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
--++
-+
-+
+-
-
Step 4.1.6.1.5.11
Pull the next terms from the original dividend down into the current dividend.
-
--++
-+
-+
+-
-+
Step 4.1.6.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
--
--++
-+
-+
+-
-+
Step 4.1.6.1.5.13
Multiply the new quotient term by the divisor.
--
--++
-+
-+
+-
-+
-+
Step 4.1.6.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
--
--++
-+
-+
+-
-+
+-
Step 4.1.6.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--
--++
-+
-+
+-
-+
+-
Step 4.1.6.1.5.16
Since the remander is , the final answer is the quotient.
Step 4.1.6.1.6
Write as a set of factors.
Step 4.1.6.2
Remove unnecessary parentheses.
Step 4.1.7
Factor.
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Step 4.1.7.1
Factor out of .
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Step 4.1.7.1.1
Factor out of .
Step 4.1.7.1.2
Factor out of .
Step 4.1.7.1.3
Factor out of .
Step 4.1.7.2
Remove unnecessary parentheses.
Step 4.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.3
Set equal to and solve for .
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Step 4.3.1
Set equal to .
Step 4.3.2
Add to both sides of the equation.
Step 4.4
Set equal to and solve for .
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Step 4.4.1
Set equal to .
Step 4.4.2
Solve for .
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Step 4.4.2.1
Use the quadratic formula to find the solutions.
Step 4.4.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 4.4.2.3
Simplify.
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Step 4.4.2.3.1
Simplify the numerator.
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Step 4.4.2.3.1.1
Raise to the power of .
Step 4.4.2.3.1.2
Multiply .
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Step 4.4.2.3.1.2.1
Multiply by .
Step 4.4.2.3.1.2.2
Multiply by .
Step 4.4.2.3.1.3
Add and .
Step 4.4.2.3.1.4
Rewrite as .
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Step 4.4.2.3.1.4.1
Factor out of .
Step 4.4.2.3.1.4.2
Rewrite as .
Step 4.4.2.3.1.5
Pull terms out from under the radical.
Step 4.4.2.3.2
Multiply by .
Step 4.4.2.3.3
Simplify .
Step 4.4.2.4
The final answer is the combination of both solutions.
Step 4.5
Set equal to and solve for .
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Step 4.5.1
Set equal to .
Step 4.5.2
Solve for .
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Step 4.5.2.1
Subtract from both sides of the equation.
Step 4.5.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.5.2.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 4.5.2.3.1
First, use the positive value of the to find the first solution.
Step 4.5.2.3.2
Next, use the negative value of the to find the second solution.
Step 4.5.2.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4.6
The final solution is all the values that make true.
Step 5
Exclude the solutions that do not make true.