Question 197269
Ignore the previous post (there is a solution).



{{{1/(x-2)-4/(x^2-4) = 5}}} Start with the given equation.



{{{1/(x-2)-4/((x-2)(x+2)) = 5}}} Factor the second denominator



Take note that {{{x<>-2}}} or {{{x<>2}}} (these values cause a division by zero)



{{{cross((x-2))(x+2)(1/cross((x-2)))-cross((x-2)(x+2))(4/(cross((x-2)(x+2)))) = 5(x-2)(x+2)}}}



{{{x+2-4 = 5(x-2)(x+2)}}} Simplify



{{{x+2-4 = 5(x^2-4)}}} FOIL



{{{x+2-4 = 5x^2-20}}} Distribute



{{{x-2 = 5x^2-20}}} Combine like terms.



{{{0 = 5x^2-20-x+2}}} Get everything to one side.



{{{0 = 5x^2-x-18}}} Combine like terms.



Notice we have a quadratic in the form of {{{Ax^2+Bx+C}}} where {{{A=5}}}, {{{B=-1}}}, and {{{C=-18}}}



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(5)(-18) ))/(2(5))}}} Plug in  {{{A=5}}}, {{{B=-1}}}, and {{{C=-18}}}



{{{x = (1 +- sqrt( (-1)^2-4(5)(-18) ))/(2(5))}}} Negate {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1-4(5)(-18) ))/(2(5))}}} Square {{{-1}}} to get {{{1}}}. 



{{{x = (1 +- sqrt( 1--360 ))/(2(5))}}} Multiply {{{4(5)(-18)}}} to get {{{-360}}}



{{{x = (1 +- sqrt( 1+360 ))/(2(5))}}} Rewrite {{{sqrt(1--360)}}} as {{{sqrt(1+360)}}}



{{{x = (1 +- sqrt( 361 ))/(2(5))}}} Add {{{1}}} to {{{360}}} to get {{{361}}}



{{{x = (1 +- sqrt( 361 ))/(10)}}} Multiply {{{2}}} and {{{5}}} to get {{{10}}}. 



{{{x = (1 +- 19)/(10)}}} Take the square root of {{{361}}} to get {{{19}}}. 



{{{x = (1 + 19)/(10)}}} or {{{x = (1 - 19)/(10)}}} Break up the expression. 



{{{x = (20)/(10)}}} or {{{x =  (-18)/(10)}}} Combine like terms. 



{{{x = 2}}} or {{{x = -9/5}}} Simplify. 



So the <i>possible</i> solutions are {{{x = 2}}} or {{{x = -9/5}}} 

  

However, recall that we stated that {{{x<>2}}} (this values causes a division by zero). So {{{x = 2}}} is NOT a solution.



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Answer:



So the only solution is {{{x = -9/5}}}