What is the least common multiple for #\frac{x}{x-2}+\frac{x}{x+3}=\frac{1}{x^2+x-6}# and how do you solve the equations?

1 Answer
May 28, 2018

Answer:

See explanation

Explanation:

#(x-2)(x+3)# by FOIL (First, Outside, Inside, Last) is #x^2+3x-2x-6#
which simplifies to #x^2+x-6#. This will be your least common multiple (LCM)

Therefore you can find a common denominator in the LCM...
#x/(x-2)((x+3)/(x+3))+x/(x+3)((x-2)/(x-2))=1/(x^2+x-6)#

Simplify to get:
#(x(x+3)+x(x-2))/(x^2+x-6)=1/(x^2+x-6)#
You see the denominators are the same, so take them out.

Now you have the following -
#x(x+3)+x(x-2)=1#

Let's distribute; now we have
#x^2+3x+x^2-2x=1#
Adding like terms, #2x^2+x=1#

Make one side equal to 0 and solve quadratic.
#2x^2+x-1=0#

Based on Symbolab, the answer is #x=-1# or #x=1/2#.