Question 62876
Solve for x:
{{{6/(x-2) + 7/(x^2-4) = (x+3)/(x+2)}}} To simplify the left side, find the common denominator then add the fractions.
The common denominator can be found by first factoring the denominator of the second fraction: {{{(x^2-4) = (x-2)(x+2)}}}, so now you have:
{{{6/(x-2) + 7/((x-2)(x+2)) = (x+3)/(x+2)}}} Now multiply the top & bottom of the first fraction by (x+2) then add the fractions:
{{{(6(x+2) + 7)/(x-2)(x+2) = (x+3)/(x+2)}}} Now multiply both sides by (x+2) and cancel where appropriate.
{{{(x+2)(6(x+2)+ 7)/(x-2)(x+2) = (x+2)(x+3)/(x+2)}}} After cancelling the (x+2)'s, and a little simplifying you have:

{{{((6x+12)+7)/(x-2) = (x+3)}}} Multiply both sides by (x-2)
{{{6x+12+7 = (x+3)(x-2)}}} Simplify:
{{{6x+19 = x^2+x-6}}} 
{{{x^2-5x-25 = 0}}} Solve this quadratic equation using the quadratic formula: {{{x = (-b+-sqrt(b^2-4ac))/2a}}}

{{{x = (-(-5)+-sqrt((-5)^2-4(1)(-25)))/2(1)}}}
{{{x = (5+-sqrt(25-(-100)))/2}}}
{{{x = (5+-sqrt(125))/2}}}
{{{x = 5/2+(5sqrt(5))/2}}} = {{{(5/2)(1+sqrt(5))}}}
{{{x = 5/2-(5sqrt(5))/2}}} = {{{(5/2)(1-sqrt(5))}}}