Question 37341
<pre><font size = 3><b>Solve for t:

   1         1          3
——————— + ——————— = —————————
 t - 1     t + 2     t<sup>2</sup> + t<sup>-2</sup>

My graphing calculator says it has solution t = -1.390957657, but
that may be too hard to calculate by hand.  But let's try anyway:

First we need to get rid of the negative exponent on the right

            3
        —————————
         t<sup>2</sup> + t<sup>-2</sup>

Multiply top and bottom by t<sup>2</sup>

          3(t<sup>2</sup>)        3t<sup>2</sup>
       —————————— = —————————
       (t<sup>2</sup> + t<sup>-2</sup>)    t<sup>4</sup> + 1

   1         1         3t<sup>2</sup>
——————— + ——————— = —————————
 t - 1     t + 2     t<sup>4</sup> + 1


Multiply each fraction through by the LCD = (t - 1)(t + 2)(t<sup>4</sup> + 1)

 (t + 2)(t<sup>4</sup> + 1) + (t - 1)(t<sup>4</sup> + 1) = 3t<sup>2</sup>(t - 1)(t + 2)

 t<sup>5</sup> + t + 2t<sup>4</sup> + 2 + t<sup>5</sup> + t - t<sup>4</sup> - 1 = 3t<sup>2</sup>(t<sup>2</sup> + t - 2)

                  2t<sup>5</sup> + t<sup>4</sup> + 2t + 1 = 3t<sup>4</sup> + 3t<sup>3</sup> - 6t<sup>2</sup> 

     2t<sup>5</sup> - 2t<sup>4</sup> - 3t<sup>3</sup> + 6t<sup>2</sup> + 2t + 1 = 0

The only rational solutions possible are ±1, ±1/2, but none of
these turn out to be solutions.  So it cannot be solved by any 
methods of ordinary algebra.  However it DOES have one irrational
solution, approximately what I got with a scientific calculator,
x = -1.390957657.  However, your teacher cannot expect you to get
that answer by hand, unless you use tedious iterative methods.

Edwin
AnlytcPhil@aol.com</pre>