Question 150890

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



{{{cross((x+1))(x-1)((2)/cross((x+1)))+(x+1)(x-1)(6)=(x+1)cross((x-1))((3)/cross((x-1)))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{(x+1)(x-1)}}}. Doing this will eliminate all of the fractions.




{{{2(x-1)+6(x+1)(x-1)=3(x+1)}}} Simplify.



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



{{{2x-2+6x^2-6=3x+3}}} Distribute.



{{{2x-2+6x^2-6-3x-3=0}}} Subtract 3x from both sides. Subtract 3 from both sides.



{{{6x^2-x-11=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=6}}}, {{{b=-1}}}, and {{{c=-11}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-1) +- sqrt( (-1)^2-4(6)(-11) ))/(2(6))}}} Plug in  {{{a=6}}}, {{{b=-1}}}, and {{{c=-11}}}



{{{x = (1 +- sqrt( (-1)^2-4(6)(-11) ))/(2(6))}}} Negate {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1-4(6)(-11) ))/(2(6))}}} Square {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1--264 ))/(2(6))}}} Multiply {{{4(6)(-11)}}} to get {{{-264}}}



{{{x = (1 +- sqrt( 1+264 ))/(2(6))}}} Rewrite {{{sqrt(1--264)}}} as {{{sqrt(1+264)}}}



{{{x = (1 +- sqrt( 265 ))/(2(6))}}} Add {{{1}}} to {{{264}}} to get {{{265}}}



{{{x = (1 +- sqrt( 265 ))/(12)}}} Multiply {{{2}}} and {{{6}}} to get {{{12}}}. 



{{{x = (1+sqrt(265))/(12)}}} or {{{x = (1-sqrt(265))/(12)}}} Break up the expression.  



So our answers are {{{x = (1+sqrt(265))/(12)}}} or {{{x = (1-sqrt(265))/(12)}}} 



which approximate to {{{x=1.44}}} or {{{x=-1.273}}}