Question 240686
your equation is:


1/(x-4) + 4/(4x-4) = 5/4 


factor out the 4 in 4/(4x-4) to get:


1/(x-4) + 1/(x-1) = 5/4


your common denominator is going to be (x-4)*(x-1)


multiply 1/(x-4) * (x-1)/(x-1) to get (x-1)/((x-4)*(x-1))


multiply 1/(x-1) * (x-4)/(x-4) to get (x-4)/(x-4)*(x-1)))


your equation becomes:


(x-1)/((x-4)*(x-1)) + (x-4)/((x-4)*x-1)) = 5/4


combine the two fractions with the common denominator to get:


((x-1) + (x-4)) / ((x-4)*(x-1)) = 5/4


simplify by combining like terms to get:


(2x-5)/ ((x-4)*(x-1)) = 5/4


multiply both sides of this equation by 4 to get:


(4 * (2x-5))/ ((x-4)*(x-1)) = 5


simplify by removing parentheses in the numerator on the left hand side of the equation to get:


(8x-20) /  ((x-4)*(x-1)) = 5


multiply both sides of this equation by  ((x-4)*(x-1))  to get:


8x-20 = 5 *  ((x-4)*(x-1)) 


simplify by performing indicated operations and removing parentheses to get:


8x-20 = 5x^2 - 25x + 20


subtract 8x from both sides of the equation and add 20 to both sides of the equation to get:


0 = 5x^2 - 33x + 40 which is the same as:


5x^2 - 33x + 40 = 0


use the quadratic equation to solve this equation to get:


x = 5

or

x = 1.6


substitute for x in the original equation to confirm the result.


I did that and the results are confirmed so these values are your answer.


The quadratic formula is:


{{{-b +- sqrt(b^2-4ac)/(2a)}}}


the standard form of the quadratic equation is:


ax^2 + bx + c = 0


your equation is:


5x^2 - 33x + 40 = 0


a = 5
b = -33
c = 40


plug those values into the quadratic formula and you get:


{{{(-(-33) +- sqrt(1089 - 400))/10}}} which becomes:


{{{(33 +- sqrt(289))/10}}} which becomes:


{{{(33 +- 17)/10}}}


The result is x = 5 or x = 1.6