Question

How do you simplify the expression `(x^2 + x - 6)/(x^2 - 4) * (x^2 - 9)/( x^2 + 6x + 9)`?

Answer 1

`(x-3)/(x+2)`

Explanation:

first, factorise each expression.

`3 + -2 = 1`
`3 * -2 = -6`

`x^2 + x - 6 = (x+3)(x-2)`

`3 + 3 = 6`
`3 * 3 = 9`

`x^2 + 6x + 9 = (x+3)(x+3)`

for the other two expressions, the identity `(a+b)(a-b) = a^2-b^2` can be used.

`x^2 - 4 = x^2 - 2^2`
`= (x+2)(x-2)`

`x^2 - 9 = x^2 - 3^2`
`= (x+3)(x-3)`

putting the factorised expressions into the question gives

`((x+3)(x-2))/((x+2)(x-2)) * ((x+3)(x-3))/((x+3)(x+3))`

this can be cancelled

`((x+3)cancel((x-2)))/((x+2)cancel((x-2))) * (cancel((x+3))(x-3))/(cancel((x+3))(x+3))`

`((x+3))/((x+2)) * ((x-3))/((x+3))`

`(cancel((x+3)))/((x+2)) * ((x-3))/(cancel((x+3)))`

to give `1/(x+2) * (x-3)/1`

this is the same as `(x-3)/(x+2)`