Question 195715
{{{x/(x+4)-2/(x-1)=20/(x^2+3x-4)}}} Start with the given equation.



{{{x/(x+4)-2/(x-1)=20/((x+4)(x-1))}}} Factor. Take note that {{{x<>-4}}} or {{{x<>1}}} as these values cause a division by zero.



{{{cross((x+4))(x-1)(x/cross((x+4)))-(x+4)cross((x-1))(2/cross((x-1)))=cross((x+4)(x-1))(20/cross((x+4)(x-1)))}}} Multiply <font size="4"><b>every</b></font> term on both sides by the LCD {{{(x+4)(x-1)}}}. Doing this will eliminate all of the fractions.



{{{(x-1)(x)-(x+4)(2)=20}}} Cancel out and simplify



{{{x(x-1)-2(x+4)=20}}} Rearrange the terms.



{{{x^2-x-2x-8=20}}} Distribute.



{{{x^2-x-2x-8-20=0}}} Subtract 20 from both sides.



{{{x^2-3x-28=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-3}}}, and {{{c=-28}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-3) +- sqrt( (-3)^2-4(1)(-28) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-3}}}, and {{{c=-28}}}



{{{x = (3 +- sqrt( (-3)^2-4(1)(-28) ))/(2(1))}}} Negate {{{-3}}} to get {{{3}}}. 



{{{x = (3 +- sqrt( 9-4(1)(-28) ))/(2(1))}}} Square {{{-3}}} to get {{{9}}}. 



{{{x = (3 +- sqrt( 9--112 ))/(2(1))}}} Multiply {{{4(1)(-28)}}} to get {{{-112}}}



{{{x = (3 +- sqrt( 9+112 ))/(2(1))}}} Rewrite {{{sqrt(9--112)}}} as {{{sqrt(9+112)}}}



{{{x = (3 +- sqrt( 121 ))/(2(1))}}} Add {{{9}}} to {{{112}}} to get {{{121}}}



{{{x = (3 +- sqrt( 121 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (3 +- 11)/(2)}}} Take the square root of {{{121}}} to get {{{11}}}. 



{{{x = (3 + 11)/(2)}}} or {{{x = (3 - 11)/(2)}}} Break up the expression. 



{{{x = (14)/(2)}}} or {{{x =  (-8)/(2)}}} Combine like terms. 



{{{x = 7}}} or {{{x = -4}}} Simplify. 



So the <i>possible</i> answers are {{{x = 7}}} or {{{x = -4}}} 



However, we stated earlier that {{{x<>-4}}} or {{{x<>1}}} (they cause a division by zero). So we must ignore the <i>possible</i> solution {{{x = -4}}} (it is not in the domain)



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Answer:



So the only solution is {{{x = 7}}}