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

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.