How do you solve the equation #(x/(x-1)) + (x/3) = (5/(x-1))#?

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Answer 1 · 2018-03-13T18:13:31

#x = 2.594# or #1.927#

#x/(x-1) + x/3 = 5/(x-1)#

We need a common denominator of #(x-1) xx 3#

#3/3 xx x/(x-1) + (x-1)/(x-1) xx x/3 = 5/(x-1) xx 3/3#

#(3x)/(3x-3) + (x^2-x)/(3x-3) = 15/(3x-3)#

#(3x^2+3x - x)/(3x-3) = 15/(3x-3)#

#3x^2 + 2x = 15#

#3x^2 + 2x - 15 = 0#

#color(green)(a) = color(green)(3)#
#color(blue)(b) = color(blue)(2)#
#color(orange)(c) = color(orange)(-15)#

#-color(blue)(b) /(2 xx color(green)(a) ) +- sqrt( color(blue)(b)^2 - 4 xx color(green)(a) xx color(orange)(c))/(2 xx color(green)(a) )#

#-color(blue)(2) /(2 xx color(green)(3) ) +- sqrt( color(blue)(2)^2 - 4 xx color(green)(3) xx color(orange)(15))/(2 xx color(green)(3) )#

#-1/3 +- sqrt(4+180)/6#

#1/3 +- sqrt(184)/6#

Let's simplify the Square Root

#184 = 2 xx 2 xx 2 xx 23 = 2^2 xx 46#

#sqrt(184) = sqrt(2^2) xx sqrt(46) = 2 xx sqrt(46)#

#-1/3 +- (2sqrt(46))/6#

#-1/3 +- sqrt(46)/3#

So #x = 2.594# or #1.927#