Question 200942
{{{x/(x+1)+5/x=1/(x^2+x)}}} Start with the given equation.



{{{x/(x+1)+5/x=1/(x(x+1))}}} Factor the last denominator



{{{x*cross((x+1))(x/cross((x+1)))+cross(x)(x+1)(5/cross(x))=cross(x(x+1))(1/cross(x(x+1)))}}} Multiply EVERY term by the LCD {{{x(x+1)}}} to clear out the fractions.



{{{x*x+5(x+1)=1}}} Simplify



{{{x*x+5*x+5*1=1}}} Distribute



{{{x^2+5x+5=1}}} Multiply



{{{x^2+5x+5-1=0}}} Subtract 1 from both sides.



{{{x^2+5x+4=0}}} Combine like terms.



Notice that the quadratic {{{x^2+5x+4}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=5}}}, and {{{C=4}}}



Let's use the quadratic formula to solve for "x":



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(5) +- sqrt( (5)^2-4(1)(4) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=5}}}, and {{{C=4}}}



{{{x = (-5 +- sqrt( 25-4(1)(4) ))/(2(1))}}} Square {{{5}}} to get {{{25}}}. 



{{{x = (-5 +- sqrt( 25-16 ))/(2(1))}}} Multiply {{{4(1)(4)}}} to get {{{16}}}



{{{x = (-5 +- sqrt( 9 ))/(2(1))}}} Subtract {{{16}}} from {{{25}}} to get {{{9}}}



{{{x = (-5 +- sqrt( 9 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (-5 +- 3)/(2)}}} Take the square root of {{{9}}} to get {{{3}}}. 



{{{x = (-5 + 3)/(2)}}} or {{{x = (-5 - 3)/(2)}}} Break up the expression. 



{{{x = (-2)/(2)}}} or {{{x =  (-8)/(2)}}} Combine like terms. 



{{{x = -1}}} or {{{x = -4}}} Simplify. 



So the <i>possible</i> solutions are {{{x = -1}}} or {{{x = -4}}} 

  
  
However, recall that {{{x<>-1}}}, this means that the only solution is {{{x = -4}}} 


=============================================================


Answer:


So the solution is {{{x = -4}}}