Question 171794
{{{2=(x+3)/(x+2)+5/(x^2-x-6)}}} Start with the given equation.



{{{2=(x+3)/(x+2)+5/((x+2)(x-3))}}} Factor {{{x^2-x-6}}} to get {{{(x+2)(x-3)}}}



Take note that the LCD is {{{(x+2)(x-3)}}}




{{{2(x+2)(x-3)=((x+3)/cross((x+2)))(cross((x+2))(x-3))+(5/(cross((x+2)(x-3))))(cross((x+2)(x-3)))}}} Multiply EVERY term by the LCD {{{(x+2)(x-3)}}} to clear out the fractions.



{{{2(x+2)(x-3)=(x+3)(x-3)+5}}} Multiply and simplify



{{{2(x^2-x-6)=x^2-9+5}}} FOIL



{{{2x^2-2x-12=x^2-9+5}}} Distribute



{{{2x^2-2x-12=x^2-4}}} Combine like terms.



{{{2x^2-2x-12-x^2+4=0}}} Subtract {{{x^2}}} from both sides. Add 4 to both sides.



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




Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-2}}}, and {{{c=-8}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-2) +- sqrt( (-2)^2-4(1)(-8) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-2}}}, and {{{c=-8}}}



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



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



{{{x = (2 +- sqrt( 4--32 ))/(2(1))}}} Multiply {{{4(1)(-8)}}} to get {{{-32}}}



{{{x = (2 +- sqrt( 4+32 ))/(2(1))}}} Rewrite {{{sqrt(4--32)}}} as {{{sqrt(4+32)}}}



{{{x = (2 +- sqrt( 36 ))/(2(1))}}} Add {{{4}}} to {{{32}}} to get {{{36}}}



{{{x = (2 +- sqrt( 36 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (2 +- 6)/(2)}}} Take the square root of {{{36}}} to get {{{6}}}. 



{{{x = (2 + 6)/(2)}}} or {{{x = (2 - 6)/(2)}}} Break up the expression. 



{{{x = (8)/(2)}}} or {{{x =  (-4)/(2)}}} Combine like terms. 



{{{x = 4}}} or {{{x = -2}}} Simplify. 



So the possible solutions are {{{x = 4}}} or {{{x = -2}}} 

  

However, if you plug in {{{x = -2}}} back into the original equation, a division by zero will occur. 


So {{{x = -2}}} is NOT a solution.



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Answer:



So the solution is {{{x = 4}}}