Question 185493
Is the equation {{{3/x+1=3/(x-1)}}} ??? (parenthesis are critical in this case)



If so, then....



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



Take note that the LCD is {{{x(x-1)}}}



{{{cross(x)(x-1)(3/cross(x))+x(x-1)(1)=x*cross((x-1))(3/cross((x-1)))}}} Multiply EVERY term by the LCD to clear out the fractions.




{{{x-1+x(x-1)=3x}}} Simplify



{{{x-1+x^2-x=3x}}} Distribute



{{{x-1+x^2-x-3x=0}}} Subtract 3x from both sides.



{{{x^2-3x-1=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-3}}}, and {{{c=-1}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-3) +- sqrt( (-3)^2-4(1)(-1) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-3}}}, and {{{c=-1}}}



{{{x = (3 +- sqrt( (-3)^2-4(1)(-1) ))/(2(1))}}} Negate {{{-3}}} to get {{{3}}}. 



{{{x = (3 +- sqrt( 9-4(1)(-1) ))/(2(1))}}} Square {{{-3}}} to get {{{9}}}. 



{{{x = (3 +- sqrt( 9--4 ))/(2(1))}}} Multiply {{{4(1)(-1)}}} to get {{{-4}}}



{{{x = (3 +- sqrt( 9+4 ))/(2(1))}}} Rewrite {{{sqrt(9--4)}}} as {{{sqrt(9+4)}}}



{{{x = (3 +- sqrt( 13 ))/(2(1))}}} Add {{{9}}} to {{{4}}} to get {{{13}}}



{{{x = (3 +- sqrt( 13 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (3+sqrt(13))/(2)}}} or {{{x = (3-sqrt(13))/(2)}}} Break up the expression.  



So the answers are {{{x = (3+sqrt(13))/(2)}}} or {{{x = (3-sqrt(13))/(2)}}} 



which approximate to {{{x=3.303}}} or {{{x=-0.303}}}