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

2 Answers
Apr 9, 2016

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!

Apr 9, 2016

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#