How do you simplify #(x^2-9)/(x^2-16)*(x^2-8x+16)/(x^2+6x+9)#?

2 Answers
George C.
Jul 23, 2015

Factor and cancel out common factors to find:

#(x^2-9)/(x^2-16)*(x^2-8x+16)/(x^2+6x+9)#

#=((x-3)(x-4))/((x+3)(x+4))#

#=1-(14x)/((x+3)(x+4))#

with exclusion #x != 4#

Explanation:

Use difference of squares identity:

#a^2-b^2 = (a-b)(a+b)#

Use perfect square trinomial identity:

#(a+b)^2 = a^2+2ab+b^2#

#(x^2-9)/(x^2-16)*(x^2-8x+16)/(x^2+6x+9)#

#=((x^2-3^2)(x^2-2*4x+4^2))/((x^2-4^2)(x^2+2*3x+3^2))#

#=((x-3)(x+3)(x-4)^2)/((x-4)(x+4)(x+3)^2)#

#=((x-3)(x-4))/((x+3)(x+4))#

#=(x^2-7x+12)/(x^2+7x+12)#

#=((x^2+7x+12)-14x)/(x^2+7x+12)#

#=1-(14x)/((x+3)(x+4))#

with exclusion #x != 4#

Sudip Sinha
Jul 23, 2015

# ( (x - 3) (x - 4) ) / ( (x + 3) (x + 4) ) #

Explanation:

# (x^2 - 9) / (x^2 - 16) . (x^2 - 8x + 16) / (x^2 + 6x + 9) #
# = ( (x + 3) (x - 3) ) / ( (x + 4) (x - 4) ) . (x - 4)^2 / (x + 3)^2 #
# = ( (x - 3) (x - 4) ) / ( (x + 3) (x + 4) ) #

Ideas:

  1. # (x + y)^2 = x^2 + 2xy + y^2 #
    Thus, in the numerator in the fraction on the right, put #y=4# and in the denominator, put #y=3#.

  2. # x^2 - y^2 = (x + y) (x - y) #
    Again, like in the previous point, in the numerator in the fraction on the left, put #y=3# and in the denominator, put #y=4#.

  3. In the 3rd step, cancel out common terms, namely #(x-4)# and #(x+3)#.

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