How do you simplify the expression #(x+6)/(x^2-4) - (x-3)/(x+2) + (x-3)/(x-2)#?

Question

How do you simplify the expression #(x+6)/(x^2-4)  -  (x-3)/(x+2)  +  (x-3)/(x-2)#?

Nathan Axel · Accepted Answer

#(5x-6)/[(x+2)(x-2)]= (5x-6)/[(x^2+4)#

Rewrite #x^2-4# as #x^2-2^2#

#(x+6)/(x^2−2^2)-(x−3)/(x+2)+(x−3)/(x−2)#

Since #color(red)(a^2-b^2=(a+b) (a-b),#

#= (x+6)/color(red)[(x+2)(x-2)]-(x-3)/(x+2)+(x-3)/(x-2)#

Taking the LCM

#= [(x+6)-(x-3)(x-2)+(x-3)(x+2)]/[(x+2)(x-2)]#

Expand

#=[x+6-x^2+2x+3x-6+x^2+2x-3x-6]/[(x+2)(x-2)]#

Collect like terms

#= [(x+2x+3x+2x-3x)+(6-6-6)+(-x^2+x)]/[(x+2)(x-2)]#

Simplify

#= (5x-6)/[(x+2)(x-2)] = (5x-6)/[(x^2+4)#

~Hope this helps! :)

Abigail Allen · Answer

undefined To simplify the expression (x+6)/(x^2-4) - (x-3)/(x+2) + (x-3)/(x-2), we need to find a common denominator for all the fractions. The common denominator is (x^2-4)(x+2)(x-2).

Next, we multiply each fraction by the appropriate factor to obtain the common denominator.

For the first fraction (x+6)/(x^2-4), we multiply the numerator and denominator by (x+2)(x-2).

For the second fraction (x-3)/(x+2), we multiply the numerator and denominator by (x^2-4).

For the third fraction (x-3)/(x-2), we multiply the numerator and denominator by (x+2).

After multiplying, we can combine the numerators over the common denominator.

The simplified expression is (x^2 - 3x + 6)/(x^3 - 4x^2 - 4x + 16).