document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117462>Question 1117462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Demonstrate that the roots of the following equation are rational\r<BR>\n" );
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document.write( "abc²x² + 3a²cx + b²cx - 6a² - ab + 2b² = 0<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732589 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=732589\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Part of my solution to the problem is basically the same as the other tutor's; the other part is far different.<br><BR>\n" );
document.write( "As he says, put the equation in pure quadratic form:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28abc%5E2%29x%5E2%2B%283a%5E2c%2Bb%5E2c%29x%2B%28-6a%5E2-ab%2B2b%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%28abc%5E2%29x%5E2%2B%283a%5E2c%2Bb%5E2c%29x%2B%28-6a%5E2-ab%2B2b%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In a quadratic equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=px%5E2%2Bqx%2Br=0 ALIGN=MIDDLE  ALT=\"px%5E2%2Bqx%2Br=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "the sum of the roots is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-q%2Fp ALIGN=MIDDLE  ALT=\"-q%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the product of the roots is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Fp ALIGN=MIDDLE  ALT=\"r%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "In this equation, the product of the roots is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6a%5E2-ab%2B2b%5E2%29%2F%28abc%5E2%29 ALIGN=MIDDLE  ALT=\"-6a%5E2-ab%2B2b%5E2%29%2F%28abc%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-%283a%2B2b%29%282a-b%29%29%2F%28abc%5E2%29 ALIGN=MIDDLE  ALT=\"%28-%283a%2B2b%29%282a-b%29%29%2F%28abc%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With abc^2 as the denominator in the product of the two roots, the denominators of the two roots are most likely ac and bc.  There are then four possible pairs of roots:<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283a%2B2b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"-%283a%2B2b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-b%29%2F%28bc%29 ALIGN=MIDDLE  ALT=\"%282a-b%29%2F%28bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283a%2B2b%29%2F%28bc%29 ALIGN=MIDDLE  ALT=\"-%283a%2B2b%29%2F%28bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"%282a-b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283a%2B2b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"%283a%2B2b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%282a-b%29%2F%28bc%29 ALIGN=MIDDLE  ALT=\"-%282a-b%29%2F%28bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(4) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283a%2B2b%29%2F%28bc%29 ALIGN=MIDDLE  ALT=\"%283a%2B2b%29%2F%28bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%282a-b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"-%282a-b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find which is the correct pair of roots, we use the fact that the sum of the roots has to be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3a%5E2c-b%5E2c%29%2F%28abc%5E2%29+=+%28-3a%5E2-b%5E2%29%2F%28abc%29 ALIGN=MIDDLE  ALT=\"%28-3a%5E2c-b%5E2c%29%2F%28abc%5E2%29+=+%28-3a%5E2-b%5E2%29%2F%28abc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The second pair of roots gives us this sum:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283a%2B2b%29%2F%28bc%29%2B%282a-b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"-%283a%2B2b%29%2F%28bc%29%2B%282a-b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3a%5E2-2ab%2B2ab-b%5E2%29%2F%28abc%29 ALIGN=MIDDLE  ALT=\"%28-3a%5E2-2ab%2B2ab-b%5E2%29%2F%28abc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3a%5E2-b%5E2%29%2F%28abc%29 ALIGN=MIDDLE  ALT=\"%28-3a%5E2-b%5E2%29%2F%28abc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the two roots are<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283a%2B2b%29%2F%28bc%29 ALIGN=MIDDLE  ALT=\"-%283a%2B2b%29%2F%28bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282a-b%29%2F%28ac%29 ALIGN=MIDDLE  ALT=\"%282a-b%29%2F%28ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And as long as none of a, b, or c is 0, those roots are rational. </SPAN>\n" );
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