Question 1086139
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I walked in the park and thought on this problem.


I got two solutions. Each is as short as 2 - 4 lines.




<U>Solution 1</U> &nbsp;&nbsp;(the Vieta's theorem; &nbsp;&nbsp;<U>4-lines solution</U>)


<pre>
The Vieta's theorem says:  if  p and q are the roots of a quadratic equation  {{{x^2 + ux + v}}} = 0  then  u = -(p+q)  and  v = pq.

The opposite is also TRUE:  if  u = -(p+q)  and  v = pq  then p and q  are the roots of the quadratic equation    {{{x^2 + ux + v}}} = 0.


Now look into your equation and notice that  the numbers  -a  and  {{{(1/a)}}}  give  {{{-(a - (1/a))}}} when summed up and -1 when multiplied.


Hence,  the numbers "-a" and  {{{1/a}}} are the roots of your equation.
</pre>


<U>Solution 2</U> &nbsp;&nbsp;(Factoring.  &nbsp;&nbsp;<U>2-lines solution</U>)


<pre>
Factor your polynomial:  {{{x^2 + (a-(1/a))x - 1 }}} = {{{(x+a)*(x-1/a)}}}.


Hence, the roots are "-a" and {{{1/a}}}.
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