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 $1.927$

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