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

2 Answers
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

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).