Question 165095
{{{4+3/(x+2)=4/(x^2+2x)}}} Start with the given equation



{{{4+3/(x+2)=4/(x(x+2))}}} Factor {{{x^2+2x}}} to get {{{x(x+2)}}}



So the LCD is {{{x(x+2)}}}



{{{4(x(x+2))+(3/cross(x+2))(x*cross((x+2)))=(4/cross((x(x+2))))cross((x(x+2)))}}} Multiply EVERY term by the LCD {{{x(x+2)}}} to clear the fractions.



{{{4x(x+2)+3x=4}}} Simplify.



{{{4x^2+8x+3x=4}}} Distribute.



{{{4x^2+8x+3x-4=0}}} Subtract 4 from both sides.



{{{4x^2+11x-4=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=11}}}, and {{{c=-4}}}



Let's use the quadratic formula to solve for x



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



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



{{{x = (-11 +- sqrt( 121-4(4)(-4) ))/(2(4))}}} Square {{{11}}} to get {{{121}}}. 



{{{x = (-11 +- sqrt( 121--64 ))/(2(4))}}} Multiply {{{4(4)(-4)}}} to get {{{-64}}}



{{{x = (-11 +- sqrt( 121+64 ))/(2(4))}}} Rewrite {{{sqrt(121--64)}}} as {{{sqrt(121+64)}}}



{{{x = (-11 +- sqrt( 185 ))/(2(4))}}} Add {{{121}}} to {{{64}}} to get {{{185}}}



{{{x = (-11 +- sqrt( 185 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (-11+sqrt(185))/(8)}}} or {{{x = (-11-sqrt(185))/(8)}}} Break up the expression.  



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Answer:



So the solutions are {{{x = (-11+sqrt(185))/(8)}}} or {{{x = (-11-sqrt(185))/(8)}}} 



which approximate to {{{x=0.325}}} or {{{x=-3.075}}}