How do you solve x/(x-5) - 2/(x+5) = 50/(x^2-25)xx−5−2x+5=50x2−25?

2 Answers
David Britz
Apr 9, 2016

Start by multiplying both sides by the Least Common Denominator.

Explanation:

x/(x-5)-2/(x+5)=50/(x^2-25)xx−5−2x+5=50x2−25

The Least Common Denominator is (x+5)(x-5)(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+5)(x−5)

x(x+5)-2(x-5)=50x(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=50x2+5x−2x+10=50
x^2+3x-40=0x2+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(x+8)(x−5)=0
Solve each factor for 0: x={-8,5}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=-8x=−8

Hope this helps!

P dilip_k
Apr 9, 2016

Multiplying both sides by x^2-25x2−25 we get

(x(x^2-25))/(x-5)-(2(x^2-25))/(x+5)=(50(x^2-25))/(x^2-25)x(x2−25)x−5−2(x2−25)x+5=50(x2−25)x2−25
=>x(x+5)-2(x-5)=50⇒x(x+5)−2(x−5)=50
=>x^2+5x-2x+10-50=0⇒x2+5x−2x+10−50=0
=>x^2+8x-5x-40=0⇒x2+8x−5x−40=0
=>x(x+8)-5(x+8)⇒x(x+8)−5(x+8)
=>(x+8)(x-5)⇒(x+8)(x−5)
:. x=-8 and x=5

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