Question

How do you solve `x/(x-5) - 2/(x+5) = 50/(x^2-25)`?

Answer 1

Start by multiplying both sides by the Least Common Denominator.

Explanation:

`x/(x-5)-2/(x+5)=50/(x^2-25)`

The Least Common Denominator is `(x+5)(x-5)` since that is the simplest expression that all denominators will divide into. So all three terms get multiplied by `(x+5)(x-5)`

`x(x+5)-2(x-5)=50`
Above we have multiplied all numerators by the LCD and canceled out terms with the denominator where needed.

Now distribute and add like terms.
`x^2+5x-2x+10=50`
`x^2+3x-40=0`
Note that we need to solve for 0 in order to solve by factoring.

Factor the left side: `(x+8)(x-5)=0`
Solve each factor for 0: `x={-8,5}`

HOWEVER the second answer must be eliminated since it would result in a zero denominator (extraneous solution).
So the only valid answer is `x=-8`

Hope this helps!

Answer 2

Multiplying both sides by `x^2-25` we get

`(x(x^2-25))/(x-5)-(2(x^2-25))/(x+5)=(50(x^2-25))/(x^2-25)`
`=>x(x+5)-2(x-5)=50`
`=>x^2+5x-2x+10-50=0`
`=>x^2+8x-5x-40=0`
`=>x(x+8)-5(x+8)`
`=>(x+8)(x-5)`
`:. x=-8 and x=5`