How do you simplify the expression (x^2 + x - 6)/(x^2 - 4) * (x^2 - 9)/( x^2 + 6x + 9)x2+x6x24x29x2+6x+9?

1 Answer
Mar 11, 2018

(x-3)/(x+2)x3x+2

Explanation:

first, factorise each expression.

3 + -2 = 13+2=1
3 * -2 = -632=6

x^2 + x - 6 = (x+3)(x-2)x2+x6=(x+3)(x2)

3 + 3 = 63+3=6
3 * 3 = 933=9

x^2 + 6x + 9 = (x+3)(x+3)x2+6x+9=(x+3)(x+3)

for the other two expressions, the identity (a+b)(a-b) = a^2-b^2(a+b)(ab)=a2b2 can be used.

x^2 - 4 = x^2 - 2^2x24=x222
= (x+2)(x-2)=(x+2)(x2)

x^2 - 9 = x^2 - 3^2x29=x232
= (x+3)(x-3)=(x+3)(x3)

putting the factorised expressions into the question gives

((x+3)(x-2))/((x+2)(x-2)) * ((x+3)(x-3))/((x+3)(x+3))(x+3)(x2)(x+2)(x2)(x+3)(x3)(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)