Question 394345
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You want to multiply both sides of the equation by 1 in the form of the lowest common denominator divided by itself.


Since the three denominators have no common factors, the lowest common denominator is simply the product of the three denominators.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{14}{x^2}\ +\ \frac{1}{x\ -\ 1}\right)\left(\frac{(x^2)(x\ -\ 1)(x\ +\ 1)}{(x^2)(x\ -\ 1)(x\ +\ 1)}\right)\ =\ \left(\frac{3}{x\ +\ 1}\right)\left(\frac{(x^2)(x\ -\ 1)(x\ +\ 1)}{(x^2)(x\ -\ 1)(x\ +\ 1)}\right)]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{14(x\ -\ 1)(x\ +\ 1)\ +\ (x^2)(x\ +\ 1)}{(x^2)(x\ -\ 1)(x\ +\ 1)}\ =\ \frac{3(x^2)(x\ -\ 1)}{(x^2)(x\ -\ 1)(x\ +\ 1)}]


Now distribute, expand, and collect like terms in the numerators (verification is left as an exercise for the student -- and you had better verify it for yourself as I can and do make mistakes):


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{-2x^3\ +\ 18x^2\ -\ 14}{(x^2)(x\ -\ 1)(x\ +\ 1)}\ =\ 0]


Now just solve the cubic.  It doesn't factor over the rationals, so you are going to either have to use the general solution of a cubic:


Check out:


<a href="http://http://en.wikipedia.org/wiki/Cubic_function
">Wikipedia Cubic Function</a>


Or


<a href="http://http://mathworld.wolfram.com/CubicFormula.html">Wolfram Mathworld Cubic Formula</a>



But be prepared for much wailing, gnashing of teeth, and pulling of hair.


Or


Google any one of several cubic calculators.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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