Question 277025
{{{x/(x+4) - 2 = 12/(x-12)}}} Start with the given equation



{{{cross((x+4))(x-12)(x/cross((x+4))) - 2(x+4)(x-12) = (x+4)cross((x-12))(12/cross((x-12)))}}} Multiply EVERY term by the LCD {{{(x+4)(x-12)}}} to clear out the fractions.



{{{(x-12)(x)- 2(x+4)(x-12)=(x+4)(12)}}} Simplify 



{{{x(x-12)- 2(x+4)(x-12)=12(x+4)}}} Rearrange the terms.



{{{x(x-12)- 2(x^2-8x-48)=12(x+4)}}} FOIL



{{{x^2-12x- 2x^2+16x+96=12x+48}}} Distribute



{{{x^2-12x-2x^2+16x+96-12x-48=0}}} Get every term to the left side.



{{{-x^2-8x+48=0}}} Combine like terms.



Notice that the quadratic {{{-x^2-8x+48}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=-1}}}, {{{B=-8}}}, and {{{C=48}}}



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



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



{{{x = (-(-8) +- sqrt( (-8)^2-4(-1)(48) ))/(2(-1))}}} Plug in  {{{A=-1}}}, {{{B=-8}}}, and {{{C=48}}}



{{{x = (8 +- sqrt( (-8)^2-4(-1)(48) ))/(2(-1))}}} Negate {{{-8}}} to get {{{8}}}. 



{{{x = (8 +- sqrt( 64-4(-1)(48) ))/(2(-1))}}} Square {{{-8}}} to get {{{64}}}. 



{{{x = (8 +- sqrt( 64--192 ))/(2(-1))}}} Multiply {{{4(-1)(48)}}} to get {{{-192}}}



{{{x = (8 +- sqrt( 64+192 ))/(2(-1))}}} Rewrite {{{sqrt(64--192)}}} as {{{sqrt(64+192)}}}



{{{x = (8 +- sqrt( 256 ))/(2(-1))}}} Add {{{64}}} to {{{192}}} to get {{{256}}}



{{{x = (8 +- sqrt( 256 ))/(-2)}}} Multiply {{{2}}} and {{{-1}}} to get {{{-2}}}. 



{{{x = (8 +- 16)/(-2)}}} Take the square root of {{{256}}} to get {{{16}}}. 



{{{x = (8 + 16)/(-2)}}} or {{{x = (8 - 16)/(-2)}}} Break up the expression. 



{{{x = (24)/(-2)}}} or {{{x =  (-8)/(-2)}}} Combine like terms. 



{{{x = -12}}} or {{{x = 4}}} Simplify. 



So the solutions are {{{x = -12}}} or {{{x = 4}}}



I'll let you check these solutions.