Question 1086799
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<pre>
If {{{x^2 + 1/x^2}}} = 102,  then  {{{x^2}}} - 2 + {{{1/x^2)}}} = 102 - 2 = 100, 

or  {{{(x-1/x)^2}}} = 100,  which implies   {{{x - 1/x}}} = 10  or  {{{x-1/x}}} = -10.


If  {{{x - 1/x}}} = 10  then

    {{{(x - 1/x)^3}}} = 1000 = {{{x^3 - 3x^2*(1/x) + 3x*(1/x^2) - 1/x^3}}} = {{{(x^3 -1/x^3)}}} - {{{(3x - 3/x)}}} = {{{(x^3 - 1/x^3)}}} - {{{3*(x-1/x)}}} = {{{(x^3 - 1/x^3)}}} - 3*10.


It implies  {{{(x^3 - 1/x^3)}}} = 1030.



If  {{{x - 1/x}}} = -10  then

    {{{(x - 1/x)^3}}} = -1000 = {{{x^3 - 3x^2*(1/x) + 3x*(1/x^2) - 1/x^3}}} = {{{(x^3 -1/x^3)}}} - {{{3x - 3/x)}}} = {{{(x^3 - 1/x^3)}}} - {{{3*(x-1/x)}}} = {{{(x^3 - 1/x^3)}}} - 3*(-10) = {{{(x^3 - 1/x^3)}}} + 30.


It implies  {{{(x^3 - 1/x^3)}}} = -1000 - 30 = -1030.
</pre>

<U>Answer</U>. &nbsp;Under the given condition, &nbsp;the expression &nbsp;{{{(x^3 - 1/x^3)}}}&nbsp; may have two values: &nbsp;1030  &nbsp;or  &nbsp;-1030.



Solved.



<U>Two lessons to learn from this solution</U>


&nbsp;&nbsp;&nbsp;&nbsp;1. &nbsp;You do not need to solve equations to get the answer.


&nbsp;&nbsp;&nbsp;&nbsp;2. &nbsp;The approach the tutor @rothauserc took to solve the problem is <U>WRONG</U>.