Question

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

Answer 1

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`

Answer 2

`( (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)`.