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

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Answer 1 · 2018-03-11T12:37:55

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

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)#