Solve for y (y+5)/(y^2-2y)-14/(y^2-4)=0

Math
y+5y2-2y-14y2-4=0
Factor each term.
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Factor y out of y2-2y.
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Factor y out of y2.
y+5y⋅y-2y-14y2-4=0
Factor y out of -2y.
y+5y⋅y+y⋅-2-14y2-4=0
Factor y out of y⋅y+y⋅-2.
y+5y(y-2)-14y2-4=0
y+5y(y-2)-14y2-4=0
Rewrite 4 as 22.
y+5y(y-2)-14y2-22=0
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=2.
y+5y(y-2)-14(y+2)(y-2)=0
y+5y(y-2)-14(y+2)(y-2)=0
Find the LCD of the terms in the equation.
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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
y(y-2),(y+2)(y-2),1
Since y(y-2),(y+2)(y-2),1 contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.
Steps to find the LCM for y(y-2),(y+2)(y-2),1 are:
1. Find the LCM for the numeric part 1,1,1.
2. Find the LCM for the variable part y1.
3. Find the LCM for the compound variable part y-2,y+2,y-2.
4. Multiply each LCM together.
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 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for y1 is y itself.
y1=y
y occurs 1 time.
The LCM of y1 is the result of multiplying all prime factors the greatest number of times they occur in either term.
y
The factor for y-2 is y-2 itself.
(y-2)=y-2
(y-2) occurs 1 time.
The factor for y+2 is y+2 itself.
(y+2)=y+2
(y+2) occurs 1 time.
The factor for y-2 is y-2 itself.
(y-2)=y-2
(y-2) occurs 1 time.
The LCM of y-2,y+2,y-2 is the result of multiplying all factors the greatest number of times they occur in either term.
(y-2)(y+2)
The Least Common Multiple LCM of some numbers is the smallest number that the numbers are factors of.
y(y-2)(y+2)
y(y-2)(y+2)
Multiply each term by y(y-2)(y+2) and simplify.
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Multiply each term in y+5y(y-2)-14(y+2)(y-2)=0 by y(y-2)(y+2) in order to remove all the denominators from the equation.
y+5y(y-2)⋅(y(y-2)(y+2))-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Simplify y+5y(y-2)⋅(y(y-2)(y+2))-14(y+2)(y-2)⋅(y(y-2)(y+2)).
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Simplify each term.
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Cancel the common factor of y(y-2).
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Cancel the common factor.
y+5y(y-2)⋅(y(y-2)(y+2))-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Rewrite the expression.
(y+5)⋅(y+2)-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
(y+5)⋅(y+2)-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Expand (y+5)(y+2) using the FOIL Method.
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Apply the distributive property.
y(y+2)+5(y+2)-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Apply the distributive property.
y⋅y+y⋅2+5(y+2)-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Apply the distributive property.
y⋅y+y⋅2+5y+5⋅2-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
y⋅y+y⋅2+5y+5⋅2-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Simplify and combine like terms.
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Simplify each term.
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Multiply y by y.
y2+y⋅2+5y+5⋅2-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Move 2 to the left of y.
y2+2⋅y+5y+5⋅2-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Multiply 5 by 2.
y2+2y+5y+10-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
y2+2y+5y+10-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Add 2y and 5y.
y2+7y+10-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
y2+7y+10-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Cancel the common factor of (y-2)(y+2).
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Move the leading negative in -14(y+2)(y-2) into the numerator.
y2+7y+10+-14(y+2)(y-2)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Factor (y-2)(y+2) out of (y+2)(y-2).
y2+7y+10+-14(y-2)(y+2)(1)⋅(y(y-2)(y+2))=0⋅(y(y-2)(y+2))
Factor (y-2)(y+2) out of y(y-2)(y+2).
y2+7y+10+-14(y-2)(y+2)(1)⋅((y-2)(y+2)(y))=0⋅(y(y-2)(y+2))
Cancel the common factor.
y2+7y+10+-14(y-2)(y+2)⋅1⋅((y-2)(y+2)y)=0⋅(y(y-2)(y+2))
Rewrite the expression.
y2+7y+10-14⋅y=0⋅(y(y-2)(y+2))
y2+7y+10-14y=0⋅(y(y-2)(y+2))
y2+7y+10-14y=0⋅(y(y-2)(y+2))
Subtract 14y from 7y.
y2-7y+10=0⋅(y(y-2)(y+2))
y2-7y+10=0⋅(y(y-2)(y+2))
Simplify 0⋅(y(y-2)(y+2)).
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Simplify by multiplying through.
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Apply the distributive property.
y2-7y+10=0⋅((y⋅y+y⋅-2)(y+2))
Simplify the expression.
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Multiply y by y.
y2-7y+10=0⋅((y2+y⋅-2)(y+2))
Move -2 to the left of y.
y2-7y+10=0⋅((y2-2⋅y)(y+2))
y2-7y+10=0⋅((y2-2⋅y)(y+2))
y2-7y+10=0⋅((y2-2⋅y)(y+2))
Expand (y2-2y)(y+2) using the FOIL Method.
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Apply the distributive property.
y2-7y+10=0⋅(y2(y+2)-2y(y+2))
Apply the distributive property.
y2-7y+10=0⋅(y2y+y2⋅2-2y(y+2))
Apply the distributive property.
y2-7y+10=0⋅(y2y+y2⋅2-2y⋅y-2y⋅2)
y2-7y+10=0⋅(y2y+y2⋅2-2y⋅y-2y⋅2)
Simplify and combine like terms.
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Simplify each term.
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Multiply y2 by y by adding the exponents.
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Multiply y2 by y.
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Raise y to the power of 1.
y2-7y+10=0⋅(y2y1+y2⋅2-2y⋅y-2y⋅2)
Use the power rule aman=am+n to combine exponents.
y2-7y+10=0⋅(y2+1+y2⋅2-2y⋅y-2y⋅2)
y2-7y+10=0⋅(y2+1+y2⋅2-2y⋅y-2y⋅2)
Add 2 and 1.
y2-7y+10=0⋅(y3+y2⋅2-2y⋅y-2y⋅2)
y2-7y+10=0⋅(y3+y2⋅2-2y⋅y-2y⋅2)
Move 2 to the left of y2.
y2-7y+10=0⋅(y3+2⋅y2-2y⋅y-2y⋅2)
Multiply y by y by adding the exponents.
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Move y.
y2-7y+10=0⋅(y3+2y2-2(y⋅y)-2y⋅2)
Multiply y by y.
y2-7y+10=0⋅(y3+2y2-2y2-2y⋅2)
y2-7y+10=0⋅(y3+2y2-2y2-2y⋅2)
Multiply 2 by -2.
y2-7y+10=0⋅(y3+2y2-2y2-4y)
y2-7y+10=0⋅(y3+2y2-2y2-4y)
Subtract 2y2 from 2y2.
y2-7y+10=0⋅(y3+0-4y)
Add y3 and 0.
y2-7y+10=0⋅(y3-4y)
y2-7y+10=0⋅(y3-4y)
Multiply 0 by y3-4y.
y2-7y+10=0
y2-7y+10=0
y2-7y+10=0
Solve the equation.
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Factor y2-7y+10 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 10 and whose sum is -7.
-5,-2
Write the factored form using these integers.
(y-5)(y-2)=0
(y-5)(y-2)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
y-5=0
y-2=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
y-5=0
Add 5 to both sides of the equation.
y=5
y=5
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
y-2=0
Add 2 to both sides of the equation.
y=2
y=2
The final solution is all the values that make (y-5)(y-2)=0 true.
y=5,2
y=5,2
Exclude the solutions that do not make y+5y2-2y-14y2-4=0 true.
y=5
Solve for y (y+5)/(y^2-2y)-14/(y^2-4)=0

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