How do you simplify (x^2-9)/(x^2-16)*(x^2-8x+16)/(x^2+6x+9)x2−9x2−16⋅x2−8x+16x2+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)x2−9x2−16⋅x2−8x+16x2+6x+9

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

=1-(14x)/((x+3)(x+4))=1−14x(x+3)(x+4)

with exclusion x != 4x≠4

Explanation:

Use difference of squares identity:

a^2-b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b)

Use perfect square trinomial identity:

(a+b)^2 = a^2+2ab+b^2(a+b)2=a2+2ab+b2

(x^2-9)/(x^2-16)*(x^2-8x+16)/(x^2+6x+9)x2−9x2−16⋅x2−8x+16x2+6x+9

=((x^2-3^2)(x^2-2*4x+4^2))/((x^2-4^2)(x^2+2*3x+3^2))=(x2−32)(x2−2⋅4x+42)(x2−42)(x2+2⋅3x+32)

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

=(x^2-7x+12)/(x^2+7x+12)=x2−7x+12x2+7x+12

=((x^2+7x+12)-14x)/(x^2+7x+12)=(x2+7x+12)−14xx2+7x+12

=1-(14x)/((x+3)(x+4))=1−14x(x+3)(x+4)

with exclusion x != 4x≠4

Sudip Sinha
Jul 23, 2015

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

Explanation:

(x^2 - 9) / (x^2 - 16) . (x^2 - 8x + 16) / (x^2 + 6x + 9) x2−9x2−16.x2−8x+16x2+6x+9
= ( (x + 3) (x - 3) ) / ( (x + 4) (x - 4) ) . (x - 4)^2 / (x + 3)^2 =(x+3)(x−3)(x+4)(x−4).(x−4)2(x+3)2
= ( (x - 3) (x - 4) ) / ( (x + 3) (x + 4) ) =(x−3)(x−4)(x+3)(x+4)

Ideas:

  1. (x + y)^2 = x^2 + 2xy + y^2 (x+y)2=x2+2xy+y2
    Thus, in the numerator in the fraction on the right, put y=4y=4 and in the denominator, put y=3y=3.

  2. x^2 - y^2 = (x + y) (x - y) x2−y2=(x+y)(x−y)
    Again, like in the previous point, in the numerator in the fraction on the left, put y=3y=3 and in the denominator, put y=4y=4.

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

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