How do you divide and simplify #\frac { x ^ { 2} + 4x + 4} { x ^ { 2} - 9} \div \frac { x + 2} { x - 3}#?

1 Answer
smendyka
May 23, 2017

See a solution process below:

Explanation:

First, rewrite the expression by factoring the numerator of the fraction on the left as:

#((x + 2)(x + 2))/(x^2 - 9) -: (x + 2)/(x - 3)#

Next, rewrite the expression by factoring the denominator of the fraction on the left as:

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

Next rewrite this expression as:

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

Then, use this rule for dividing fractions to divide and simplify:

#(color(red)(a)/color(blue)(b))/(color(green)(c)/color(purple)(d)) = (color(red)(a) xx color(purple)(d))/(color(blue)(b) xx color(green)(c))#

#(color(red)((x + 2)(x + 2))/color(blue)((x + 3)(x - 3)))/(color(green)(x + 2)/color(purple)(x - 3)) => (color(red)(((x + 2)(x + 2))) xx color(purple)((x - 3)))/(color(blue)(((x + 3)(x - 3))) xx color(green)((x + 2))) =>#

#(color(red)((cancel((x + 2))(x + 2))) xx color(purple)(cancel((x - 3))))/(color(blue)(((x + 3)cancel((x - 3)))) xx color(green)(cancel((x + 2)))) =>#

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

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