document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1142422>Question 1142422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 4b² + 1/b² = 2, find 8b³ + 1/b³ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #763122 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=763122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
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document.write( "NOTE: IGNORE THIS SOLUTION!<br><BR>\n" );
document.write( "The method was sound but the math was defective.  I absent-mindedly added 2 to both sides of the original equation instead of the required 4.<br><BR>\n" );
document.write( "See the response from tutor @ikleyn for the correct solution.<br><BR>\n" );
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document.write( "This is a common type of question on competitive math tests at the high school level.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4b%5E2%2B1%2Fb%5E2+=+2 ALIGN=MIDDLE  ALT=\"4b%5E2%2B1%2Fb%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note that both terms on the left are perfect squares:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282b%29%5E2+%2B+%281%2Fb%29%5E2+=+2 ALIGN=MIDDLE  ALT=\"%282b%29%5E2+%2B+%281%2Fb%29%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now remember that in squaring a binomial x+y, you get two perfect square terms plus a middle term: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29%5E2+=+x%5E2%2B2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"%28x%2By%29%5E2+=+x%5E2%2B2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Then note that if the binomial is of the form x+1/x, the middle term in the square is a constant: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%2Fx%29%5E2+=+x%5E2%2B2%2B1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B1%2Fx%29%5E2+=+x%5E2%2B2%2B1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So, in this problem, add 2 to both sides; the left side becomes a perfect square trinomial:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282b%29%5E2+%2B+4+%2B+%281%2Fb%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"%282b%29%5E2+%2B+4+%2B+%281%2Fb%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282b%2B1%2Fb%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"%282b%2B1%2Fb%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2B1%2Fb+=+2 ALIGN=MIDDLE  ALT=\"2b%2B1%2Fb+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now note that, in the expression you are to evaluate, both terms are perfect cubes: 8b^3 = (2b)^3; 1/b^3 = (1/b)^3.<br><BR>\n" );
document.write( "You will get both of those terms, plus other terms, when you cube (2b+1/b):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282b%2B1%2Fb%29%5E3+=+%282b%29%5E3%2B3%282b%29%5E2%281%2Fb%29%2B3%282b%29%281%2Fb%29%5E2%2B%281%2Fb%29%5E3+=+8b%5E3%2B12b%2B6%2Fb%2B1%2Fb%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This expression can be simplified using the value you now know of 2b+1/b:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8b%5E3%2B12b%2B6%2B1%2Fb%5E3+=+%288b%5E3%2B1%2Fb%5E3%29%2B6%282b%2B1%2Fb%29+=+%288b%5E3%2B1%2Fb%5E3%29%2B6%282%29+=+%288b%5E3%2B1%2Fb%5E3%29%2B12 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So now you have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282b%2B1%2Fb%29%5E3+=+2%5E3+=+8+=+%288b%5E3%2B1%2Fb%5E3%29%2B12 ALIGN=MIDDLE  ALT=\"%282b%2B1%2Fb%29%5E3+=+2%5E3+=+8+=+%288b%5E3%2B1%2Fb%5E3%29%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and so<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8b%5E3%2B1%2Fb%5E3+=+8-12+=+-4 ALIGN=MIDDLE  ALT=\"8b%5E3%2B1%2Fb%5E3+=+8-12+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );