Question 701287
{{{ x/(x-5)-2/(x+5)=50/(x^2-25)}}}....common denominator for {{{x-5}}} and {{{x+5}}} {{{(x-5)(x+5)}}}


{{{ x(x+5)/((x+5)(x-5))-2(x-5)/((x-5)(x+5))=50/(x^2-25)}}}....since {{{(x-5)(x+5)=(x^2-25)}}}, we will have



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


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


{{{ (x^2+3x+5)/(x^2-25))=50/(x^2-25)}}}.....cross multiply


{{{ (x^2+3x+5)(x^2-25)=50(x^2-25)}}}


{{{x^4-25x^2+3x^3-75x+5x^2-125=50x^2-1250}}}


{{{x^4-25x^2+3x^3-75x+5x^2-125-50x^2+1250=0}}}


{{{x^4+3x^3-70x^2-75x+1125=0}}}


{{{(3x^3-75x)+x^4-70x^2+1125=0}}}


{{{3x(x^2-25)+(x^4-25x^2)-(45x^2-1125)=0}}}



{{{3x(x^2-25)+x^2(x^2-25)-45(x^2-25)=0}}}


{{{(x^2-25)(x^2+3x-45)=0}}}


{{{(x-5)(x+5)(x^2+3x-45)=0}}}


solutions:

if {{{(x-5)=0}}}....=>...{{{x=5}}}

if {{{(x+5)=0}}}....=>...{{{x=-5}}}

if {{{(x^2+3x-45)=0}}}, ...we need to use quadratic formula to find zeros


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 


{{{x = (-3 +- sqrt( 3^2-4*1*(-45) ))/(2*1) }}} 


{{{x = (-3 +- sqrt( 9+180 ))/2 }}} 


{{{x = (-3 +- sqrt( 189 ))/2 }}} 


{{{x = (-3 +- 13.75)/2 }}} 

solutions:

{{{x = (-3 + 13.75)/2 }}} 

{{{x =  10.75/2 }}}

 {{{x = 5.375 }}}

and

{{{x = (-3 - 13.75)/2 }}} 

{{{x =  -16.75/2 }}}

 {{{x = -8.375 }}}


so, your solutions are:

{{{x=5}}}

{{{x=-5}}}

{{{x = 5.375 }}}

{{{x = -8.375 }}}


{{{ graph( 600, 600, -10,10, -410, 1250,circle(-5,0,0.2), (x-5)(x+5)(x^2+3x-45)) }}}