Solve for y y/(y^2-8y+15)+y/(y^2-9)=y/(y^2-2y-15)

Math
yy2-8y+15+yy2-9=yy2-2y-15
Factor each term.
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Factor y2-8y+15 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 15 and whose sum is -8.
-5,-3
Write the factored form using these integers.
y(y-5)(y-3)+yy2-9=yy2-2y-15
y(y-5)(y-3)+yy2-9=yy2-2y-15
Rewrite 9 as 32.
y(y-5)(y-3)+yy2-32=yy2-2y-15
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=3.
y(y-5)(y-3)+y(y+3)(y-3)=yy2-2y-15
Factor y2-2y-15 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 -15 and whose sum is -2.
-5,3
Write the factored form using these integers.
y(y-5)(y-3)+y(y+3)(y-3)=y(y-5)(y+3)
y(y-5)(y-3)+y(y+3)(y-3)=y(y-5)(y+3)
y(y-5)(y-3)+y(y+3)(y-3)=y(y-5)(y+3)
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-5)(y-3),(y+3)(y-3),(y-5)(y+3)
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 y-5 is y-5 itself.
(y-5)=y-5
(y-5) occurs 1 time.
The factor for y-3 is y-3 itself.
(y-3)=y-3
(y-3) occurs 1 time.
The factor for y+3 is y+3 itself.
(y+3)=y+3
(y+3) occurs 1 time.
The factor for y-3 is y-3 itself.
(y-3)=y-3
(y-3) occurs 1 time.
The factor for y-5 is y-5 itself.
(y-5)=y-5
(y-5) occurs 1 time.
The factor for y+3 is y+3 itself.
(y+3)=y+3
(y+3) occurs 1 time.
The LCM of y-5,y-3,y+3,y-3,y-5,y+3 is the result of multiplying all factors the greatest number of times they occur in either term.
(y-5)(y-3)(y+3)
(y-5)(y-3)(y+3)
Multiply each term by (y-5)(y-3)(y+3) and simplify.
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Multiply each term in y(y-5)(y-3)+y(y+3)(y-3)=y(y-5)(y+3) by (y-5)(y-3)(y+3) in order to remove all the denominators from the equation.
y(y-5)(y-3)⋅((y-5)(y-3)(y+3))+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Simplify y(y-5)(y-3)⋅((y-5)(y-3)(y+3))+y(y+3)(y-3)⋅((y-5)(y-3)(y+3)).
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Simplify each term.
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Cancel the common factor of (y-5)(y-3).
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Cancel the common factor.
y(y-5)(y-3)⋅((y-5)(y-3)(y+3))+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Rewrite the expression.
y⋅(y+3)+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
y⋅(y+3)+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Apply the distributive property.
y⋅y+y⋅3+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Multiply y by y.
y2+y⋅3+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Move 3 to the left of y.
y2+3⋅y+y(y+3)(y-3)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Cancel the common factor of (y-3)(y+3).
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Factor (y-3)(y+3) out of (y+3)(y-3).
y2+3y+y(y-3)(y+3)(1)⋅((y-5)(y-3)(y+3))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Factor (y-3)(y+3) out of (y-5)(y-3)(y+3).
y2+3y+y(y-3)(y+3)(1)⋅((y-3)(y+3)(y-5))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Cancel the common factor.
y2+3y+y(y-3)(y+3)⋅1⋅((y-3)(y+3)(y-5))=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Rewrite the expression.
y2+3y+y⋅(y-5)=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
y2+3y+y⋅(y-5)=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Apply the distributive property.
y2+3y+y⋅y+y⋅-5=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Multiply y by y.
y2+3y+y2+y⋅-5=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Move -5 to the left of y.
y2+3y+y2-5y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
y2+3y+y2-5y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Simplify by adding terms.
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Add y2 and y2.
2y2+3y-5y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Subtract 5y from 3y.
2y2-2y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
2y2-2y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
2y2-2y=y(y-5)(y+3)⋅((y-5)(y-3)(y+3))
Simplify y(y-5)(y+3)⋅((y-5)(y-3)(y+3)).
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Cancel the common factor of (y-5)(y+3).
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Factor (y-5)(y+3) out of (y-5)(y-3)(y+3).
2y2-2y=y(y-5)(y+3)⋅((y-5)(y+3)(y-3))
Cancel the common factor.
2y2-2y=y(y-5)(y+3)⋅((y-5)(y+3)(y-3))
Rewrite the expression.
2y2-2y=y⋅(y-3)
2y2-2y=y⋅(y-3)
Apply the distributive property.
2y2-2y=y⋅y+y⋅-3
Simplify the expression.
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Multiply y by y.
2y2-2y=y2+y⋅-3
Move -3 to the left of y.
2y2-2y=y2-3y
2y2-2y=y2-3y
2y2-2y=y2-3y
2y2-2y=y2-3y
Solve the equation.
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Move all terms containing y to the left side of the equation.
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Subtract y2 from both sides of the equation.
2y2-2y-y2=-3y
Add 3y to both sides of the equation.
2y2-2y-y2+3y=0
Subtract y2 from 2y2.
y2-2y+3y=0
Add -2y and 3y.
y2+y=0
y2+y=0
Factor y out of y2+y.
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Factor y out of y2.
y⋅y+y=0
Raise y to the power of 1.
y⋅y+y=0
Factor y out of y1.
y⋅y+y⋅1=0
Factor y out of y⋅y+y⋅1.
y(y+1)=0
y(y+1)=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=0
y+1=0
Set the first factor equal to 0.
y=0
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
y+1=0
Subtract 1 from both sides of the equation.
y=-1
y=-1
The final solution is all the values that make y(y+1)=0 true.
y=0,-1
y=0,-1
Solve for y y/(y^2-8y+15)+y/(y^2-9)=y/(y^2-2y-15)

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