Question

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?

Answer 1

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`.