# Simplify ((y^2-6y+9)/(-(y^2-4)))÷((y^2-9)/(y^2-8y+12))

y2-6y+9-(y2-4)÷y2-9y2-8y+12
To divide by a fraction, multiply by its reciprocal.
y2-6y+9-(y2-4)⋅y2-8y+12y2-9
Factor using the perfect square rule.
Rewrite 9 as 32.
y2-6y+32-(y2-4)⋅y2-8y+12y2-9
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅y⋅-3
Simplify.
2ab=-6y
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=y and b=-3.
(y-3)2-(y2-4)⋅y2-8y+12y2-9
(y-3)2-(y2-4)⋅y2-8y+12y2-9
Simplify the denominator.
Rewrite 4 as 22.
(y-3)2-(y2-22)⋅y2-8y+12y2-9
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-3)2-(y+2)(y-2)⋅y2-8y+12y2-9
(y-3)2-(y+2)(y-2)⋅y2-8y+12y2-9
Move the negative in front of the fraction.
-(y-3)2(y+2)(y-2)⋅y2-8y+12y2-9
Factor y2-8y+12 using the AC method.
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 12 and whose sum is -8.
-6,-2
Write the factored form using these integers.
-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-9
-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-9
Simplify the denominator.
Rewrite 9 as 32.
-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-32
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-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)
-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)
Simplify terms.
Cancel the common factor of y-3.
Move the leading negative in -(y-3)2(y+2)(y-2) into the numerator.
-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)
Factor y-3 out of -(y-3)2.
(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)
Factor y-3 out of (y+3)(y-3).
(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y-3)(y+3)
Cancel the common factor.
(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y-3)(y+3)
Rewrite the expression.
-(y-3)(y+2)(y-2)⋅(y-6)(y-2)y+3
-(y-3)(y+2)(y-2)⋅(y-6)(y-2)y+3
Cancel the common factor of y-2.
Factor y-2 out of (y+2)(y-2).
-(y-3)(y-2)(y+2)⋅(y-6)(y-2)y+3
Factor y-2 out of (y-6)(y-2).
-(y-3)(y-2)(y+2)⋅(y-2)(y-6)y+3
Cancel the common factor.
-(y-3)(y-2)(y+2)⋅(y-2)(y-6)y+3
Rewrite the expression.
-(y-3)y+2⋅y-6y+3
-(y-3)y+2⋅y-6y+3
Multiply -(y-3)y+2 and y-6y+3.
-(y-3)(y-6)(y+2)(y+3)
Move the negative in front of the fraction.
-(y-3)(y-6)(y+2)(y+3)
-(y-3)(y-6)(y+2)(y+3)
Simplify ((y^2-6y+9)/(-(y^2-4)))÷((y^2-9)/(y^2-8y+12))

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