Question 167712
{{{(1)/(1+x)-(1)/(3-x)=(6)/(35)}}} Start with the given equation.



{{{35cross((1+x))(3-x)((1)/cross((1+x)))-35(1+x)cross((3-x))((1)/cross((3-x)))=cross(35)(1+x)(3-x)((6)/cross(35))}}} Multiply <font size="4"><b>EVERY</b></font> term on both sides by the LCD {{{35(1+x)(3-x)}}}. Doing this will eliminate all of the fractions.



{{{35(3-x)-35(1+x)=6(1+x)(3-x)}}} Multiply and simplify



{{{35(3-x)-35(1+x)=6(3+2x-x^2)}}} FOIL



{{{105-35x-35-35x=18+12x-6x^2}}} Distribute



{{{-70x+70=-6x^2+12x+18}}} Combine like terms.



{{{0=-6x^2+12x+18+70x-70}}} Get all terms to the right side.



{{{0=-6x^2+82x-52}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=-6}}}, {{{b=82}}}, and {{{c=-52}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(82) +- sqrt( (82)^2-4(-6)(-52) ))/(2(-6))}}} Plug in  {{{a=-6}}}, {{{b=82}}}, and {{{c=-52}}}



{{{x = (-82 +- sqrt( 6724-4(-6)(-52) ))/(2(-6))}}} Square {{{82}}} to get {{{6724}}}. 



{{{x = (-82 +- sqrt( 6724-1248 ))/(2(-6))}}} Multiply {{{4(-6)(-52)}}} to get {{{1248}}}



{{{x = (-82 +- sqrt( 5476 ))/(2(-6))}}} Subtract {{{1248}}} from {{{6724}}} to get {{{5476}}}



{{{x = (-82 +- sqrt( 5476 ))/(-12)}}} Multiply {{{2}}} and {{{-6}}} to get {{{-12}}}. 



{{{x = (-82 +- 74)/(-12)}}} Take the square root of {{{5476}}} to get {{{74}}}. 



{{{x = (-82 + 74)/(-12)}}} or {{{x = (-82 - 74)/(-12)}}} Break up the expression. 



{{{x = (-8)/(-12)}}} or {{{x =  (-156)/(-12)}}} Combine like terms. 



{{{x = 2/3}}} or {{{x = 13}}} Simplify. 



So the answers are {{{x = 2/3}}} or {{{x = 13}}} 


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


Answer:



So the solutions are {{{x = 2/3}}} or {{{x = 13}}}