Question 1135667
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<pre>
Let's assume that  {{{(q^2-1)/(qx)}}}  is a rational number R:


    {{{(q^2-1)/(qx)}}} = R,, where R is a rational number.


Then  x = {{{(q^2-1)/(q*R)}}}, and it is <U>rational number</U>, since the numerator and denominator are rational numbers 

(partly according to the condition and partly according to the assumption).


But we are given that x  is irrational - CONTRADICTION.
</pre>

The contradiction proves that our assumption is &nbsp;FALSE.


Hence, &nbsp;the number  &nbsp;{{{(q^2-1)/(qx)}}}  &nbsp; is <U>irrational</U>.


It is exactly what has to be proved.


The proof is completed.