Question 170424
{{{(2)/(x-2)+(x)/(x+2)=(x+6)/(x^2-4)}}} Start with the given equation.



{{{(2)/(x-2)+(x)/(x+2)=(x+6)/((x-2)(x+2))}}} Factor {{{x^2-4}}} to get {{{(x-2)(x+2)}}}



Take note that the LCD is {{{(x-2)(x+2)}}}



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



{{{2(x+2)+x(x-2)=x+6}}} Multiply and simplify



{{{2x+4+x^2-2x=x+6}}} Distribute.



{{{2x+4+x^2-2x-x-6=0}}} Subtract x from both sides. Subtract 6 from both sides.



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



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



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(1)(-2) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-1}}}, and {{{c=-2}}}



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



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



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



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



{{{x = (1 +- sqrt( 9 ))/(2(1))}}} Add {{{1}}} to {{{8}}} to get {{{9}}}



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



{{{x = (1 +- 3)/(2)}}} Take the square root of {{{9}}} to get {{{3}}}. 



{{{x = (1 + 3)/(2)}}} or {{{x = (1 - 3)/(2)}}} Break up the expression. 



{{{x = (4)/(2)}}} or {{{x =  (-2)/(2)}}} Combine like terms. 



{{{x = 2}}} or {{{x = -1}}} Simplify. 



So the possible answers are {{{x = 2}}} or {{{x = -1}}} 



However, if you plug in {{{x = 2}}}, you'll get a division by zero. So {{{x = 2}}} is not an answer (since {{{x = 2}}} is not even in the domain).




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Answer:



So the solution is {{{x = -1}}}