Question 270261
{{{x/(2x+2)+1/4=-1/(x+2)}}} Start with the given equation.



{{{x/(2(x+1))+1/4=-1/(x+2)}}} Factor the first denominator.



Take note that the LCD is {{{4(x+1)(x+2)}}}



{{{cross(4)^2cross((x+1))(x+2)(x/(cross(2)cross((x+1))))+cross(4)(x+1)(x+2)(1/cross(4))=4(x+1)cross((x+2))(-1/cross((x+2)))}}} Multiply EVERY term by the LCD {{{4(x+1)(x+2)}}} to eliminate the fractions.



{{{2(x+2)x+(x+1)(x+2)=4(x+1)(-1)}}} Simplify.



{{{2x(x+2)+(x+1)(x+2)=4(-1)(x+1)}}} Rearrange the terms.



{{{2x(x+2)+(x+1)(x+2)=-4(x+1)}}} Multiply.



{{{2x^2+4x+(x+1)(x+2)=-4x-4}}} Distribute.



{{{2x^2+4x+x^2+3x+2=-4x-4}}} FOIL.



{{{2x^2+4x+x^2+3x+2+4x+4=0}}} Get every term to one side.



{{{3x^2+11x+6=0}}} Combine like terms.



Now let's use the quadratic formula to solve {{{3x^2+11x+6=0}}}



*[invoke quadratic_formula 3,11,6, "x"]



So the solutions to the equation {{{x/(2x+2)+1/4=-1/(x+2)}}} are


{{{x=-2/3}}} or {{{x=-3}}}