# Find All Complex Solutions 1/(x-1)+1/(x+2)=5/4

Find All Complex Solutions 1/(x-1)+1/(x+2)=5/4
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
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.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
has factors of and .
Multiply by .
The factor for is itself.
occurs time.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
The Least Common Multiple of some numbers is the smallest number that the numbers are factors of.
Multiply each term in by to eliminate the fractions.
Multiply each term in by .
Simplify the left side.
Simplify each term.
Rewrite using the commutative property of multiplication.
Combine and .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Rewrite using the commutative property of multiplication.
Combine and .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Simplify by adding terms.
Subtract from .
Simplify the right side.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Move to the left of .
Rewrite as .
Multiply by .
Subtract from .
Apply the distributive property.
Multiply by .
Solve the equation.
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Subtract from .
Subtract from both sides of the equation.
Subtract from .
Factor by grouping.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Factor out of .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to and solve for .
Set equal to .
Solve for .
Subtract from both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify the right side.
Move the negative in front of the fraction.
Set equal to and solve for .
Set equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
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