Question 195562
{{{2+8/(x-5)=(x+5)/(x^2-25)}}} Start with the given equation.



{{{2+8/(x-5)=(x+5)/((x-5)(x+5))}}} Factor. Take note that {{{x<>-5}}} or {{{x<>5}}} (these values cause a division by zero)



{{{2(x-5)(x+5)+cross((x-5))(x+5)(8/cross((x-5)))=cross((x-5)(x+5))((x+5)/(cross((x-5)(x+5))))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{(x-5)(x+5)}}}. Doing this will eliminate all of the fractions.



{{{2(x-5)(x+5)+8(x+5)=x+5}}}



{{{2(x^2-25)+8(x+5)=x+5}}} FOIL



{{{2x^2-50+8x+40=x+5}}} Distribute



{{{2x^2-50+8x+40-x-5=0}}} Get all terms to the left side.



{{{2x^2+7x-15=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=7}}}, and {{{c=-15}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(7) +- sqrt( (7)^2-4(2)(-15) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=7}}}, and {{{c=-15}}}



{{{x = (-7 +- sqrt( 49-4(2)(-15) ))/(2(2))}}} Square {{{7}}} to get {{{49}}}. 



{{{x = (-7 +- sqrt( 49--120 ))/(2(2))}}} Multiply {{{4(2)(-15)}}} to get {{{-120}}}



{{{x = (-7 +- sqrt( 49+120 ))/(2(2))}}} Rewrite {{{sqrt(49--120)}}} as {{{sqrt(49+120)}}}



{{{x = (-7 +- sqrt( 169 ))/(2(2))}}} Add {{{49}}} to {{{120}}} to get {{{169}}}



{{{x = (-7 +- sqrt( 169 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (-7 +- 13)/(4)}}} Take the square root of {{{169}}} to get {{{13}}}. 



{{{x = (-7 + 13)/(4)}}} or {{{x = (-7 - 13)/(4)}}} Break up the expression. 



{{{x = (6)/(4)}}} or {{{x =  (-20)/(4)}}} Combine like terms. 



{{{x = 3/2}}} or {{{x = -5}}} Simplify. 



So the <i>possible</i> answers are {{{x = 3/2}}} or {{{x = -5}}} 



However, recall that we stated earlier that {{{x<>-5}}} (since it causes a division by zero). So {{{x=-5}}} is NOT a solution.



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Answer:



So the only solution is {{{x = 3/2}}}