document.write( "Question 1142422: If 4b² + 1/b² = 2, find 8b³ + 1/b³ \n" ); document.write( "

Algebra.Com's Answer #763133 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "If 4b² + 1/b² = 2, find 8b³ + 1/b³
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\n" ); document.write( "\n" ); document.write( " The solution by the other tutor has errors.\r
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\n" ); document.write( "\n" ); document.write( " So I came to provide a correct solution.\r
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document.write( "You are given  \r\n" );
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document.write( "     +  = 2.      (1)\r\n" );
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document.write( "It is the same as\r\n" );
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document.write( "     +  = 2.\r\n" );
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document.write( "Add 4 to both sides. You will get\r\n" );
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document.write( "     + 4 +  = 6.      (2)\r\n" );
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document.write( "The left side is  nothing else as  .  Therefore, the equation (2) takes the form\r\n" );
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document.write( "     = 6.\r\n" );
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document.write( "Take the square root from both sides of the last equation. You will get\r\n" );
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document.write( "     = +/- .     (3)\r\n" );
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document.write( "Now, \r\n" );
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document.write( "     =  +  +  +  =  +  +  +  =  +  +  +  =  +  + .\r\n" );
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document.write( "Thus the very first part of this chain of equalities is equal to its very last part\r\n" );
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document.write( "     =  +  + .\r\n" );
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document.write( "Hence, \r\n" );
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document.write( "     +  =  - .\r\n" );
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document.write( "Now the final step is to substitute expression (3) into the last equality.\r\n" );
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document.write( "Since expression (3) has the sign +/-, I will make this substitution in two lines.\r\n" );
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document.write( "    Case 1.  If   = + ,  then   +  =  -  =  -  = 0.\r\n" );
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document.write( "    Case 2.  If   = - ,  then   +  =  -  =  +  = 0.\r\n" );
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document.write( "You see that for any of the two cases the answer is 0 (zero, ZERO).\r\n" );
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document.write( "ANSWER.  If   +  = 2,  then    +  = 0.\r\n" );
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\n" ); document.write( "\n" ); document.write( "This result seems to be VERY INEXPECTED, and you might be overwhelmed / stunned by this answer, \r
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\n" ); document.write( "\n" ); document.write( "but the deep reason why it is so is that the given equation HAS NO real roots \"b\".\r
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\n" ); document.write( "\n" ); document.write( "It has ONLY COMPLEX NUMBER solutions (!)\r
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\n" ); document.write( "\n" ); document.write( "See the attached plot of the function y = $\"2x%5E2\"$ + $\"1%2Fx%5E2\"$ .\r
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document.write( "    \r\n" );
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document.write( "          Plot y =  + \r\n" );
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document.write( "The plot shows that the function  y =  +   is always greater than 2, so the given equation has no real roots.\r\n" );
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\n" ); document.write( "\n" ); document.write( "This problem is slightly ABOVE an averaged High school Math competition level, since it contains TWO UNDERWATER STONES
\n" ); document.write( "instead of standard ONE.\r
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\n" ); document.write( "\n" ); document.write( "The second underwater stone is the complexity of the roots to the given equation, which makes it difficult to a student\r
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\n" ); document.write( "\n" ); document.write( "For the standard problems / (exercises) of the High school competition level, see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO evaluate expressions involving $\"%28x+%2B+1%2Fx%29\"$ , $\"%28x%5E2%2B1%2Fx%5E2%29\"$ and $\"%28x%5E3%2B1%2Fx%5E3%29\"$ \r
\n" ); document.write( "\n" ); document.write( "    - Advanced lesson on evaluating expressions\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Evaluation, substitution\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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