Question 1142422
.
If 4b² + 1/b² = 2, find 8b³ + 1/b³
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            The solution by the other tutor has errors.


            So I came to provide a correct solution.



<pre>
You are given  


    {{{4b^2}}} + {{{1/b^2}}} = 2.      (1)


It is the same as


    {{{(2b)^2}}} + {{{1/b^2}}} = 2.


Add 4 to both sides. You will get


    {{{(2b)^2}}} + 4 + {{{b^2}}} = 6.      (2)


The left side is  nothing else as  {{{(2b + 1/b)^2}}}.  Therefore, the equation (2) takes the form


    {{{(2b + 1/b)^2}}} = 6.


Take the square root from both sides of the last equation. You will get


    {{{2b + 1/b}}} = +/- {{{sqrt(6)}}}.     (3)


Now, 


    {{{(2b  + 1/b)^3}}} = {{{(2b)^3}}} + {{{3*(2b)^2*(1/b)}}} + {{{3*(2b)*(1/b^2)}}} + {{{1/b^3}}} = {{{8b^3}}} + {{{3*4*b^2*(1/b)}}} + {{{3*(2b)*(1/b^2)}}} + {{{1/b^3}}} = {{{8b^3}}} + {{{12b}}} + {{{6/b}}} + {{{1/b^3}}} = {{{8b^3}}} + {{{6*(2b + 1/b)}}} + {{{1/b^3}}}.


Thus the very first part of this chain of equalities is equal to its very last part


    {{{(2b  + 1/b)^3}}} = {{{8b^3}}} + {{{6*(2b + 1/b)}}} + {{{1/b^3}}}.


Hence, 


    {{{8b^3}}} + {{{1/b^3}}} = {{{(2b  + 1/b)^3}}} - {{{6*(2b + 1/b)}}}.


Now the final step is to substitute expression (3) into the last equality.


Since expression (3) has the sign +/-, I will make this substitution in two lines.



    <U>Case 1</U>.  If  {{{2b + 1/b}}} = + {{{sqrt(6)}}},  then  {{{8b^3}}} + {{{1/b^3}}} = {{{(sqrt(6))^3}}} - {{{6*sqrt(6)}}} = {{{6*sqrt(6)}}} - {{{6*sqrt(6)}}} = 0.



    <U>Case 2</U>.  If  {{{2b + 1/b}}} = - {{{sqrt(6)}}},  then  {{{8b^3}}} + {{{1/b^3}}} = {{{(-sqrt(6))^3}}} - {{{6*(-sqrt(6))}}} = {{{-6*sqrt(6)}}} + {{{6*sqrt(6)}}} = 0.



You see that for any of the two cases the answer is 0 (zero, ZERO).



<U>ANSWER</U>.  If  {{{4b^2}}} + {{{1/b^2}}} = 2,  then   {{{8b^3}}} + {{{1/b^3}}} = 0.
</pre>

---------------


This result seems to be &nbsp;VERY &nbsp;INEXPECTED, &nbsp;and you might be overwhelmed / stunned by this answer, 


but the deep reason why it is so is that the given equation &nbsp;<U>HAS &nbsp;NO &nbsp;real &nbsp;roots &nbsp;"b"</U>.


It has &nbsp;<U>ONLY &nbsp;COMPLEX &nbsp;NUMBER &nbsp;solutions</U> &nbsp;(!)



See the attached plot of the function  &nbsp;&nbsp;y = {{{2x^2}}} + {{{1/x^2}}}.



<pre>
    {{{graph( 330, 330, -5, 5, -1, 10,
          2x^2 + 1/x^2
)}}}


          Plot y = {{{2x^2}}} + {{{1/x^2}}}



The plot shows that the function  y = {{{2x^2}}} + {{{1/x^2}}}  is always greater than 2, so the given equation has no real roots.
</pre>


--------------------


This problem is slightly &nbsp;ABOVE&nbsp; an averaged &nbsp;High school Math competition level, &nbsp;since it contains &nbsp;<U>TWO &nbsp;UNDERWATER &nbsp;STONES</U> 
instead of standard &nbsp;ONE.


The second &nbsp;<U>underwater stone</U>&nbsp; is the complexity of the roots to the given equation, &nbsp;which makes it difficult to a student

to believe that the obtained result / (answer) &nbsp;is correct.



For the standard problems / (exercises) of the &nbsp;High school competition level,  &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/HOW-TO-evaluate-expressions-involving-x%2Binv%28x%29-x2%2Binv%28x2%29-and-x%5E3%2Binv%28x%5E3%29.lesson>HOW TO evaluate expressions involving &nbsp;{{{(x + 1/x)}}}, &nbsp;{{{(x^2+1/x^2)}}} &nbsp;and &nbsp;{{{(x^3+1/x^3)}}}</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/evaluation/Advanced-lesson-on-evaluating-expressions.lesson>Advanced lesson on evaluating expressions</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Evaluation, substitution</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.