document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=616452>Question 616452</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For which of the following equations is it true that the sum of the roots equals the product of the roots?\r<BR>\n" );
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document.write( "A. x^2 - 4 = 0\r<BR>\n" );
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document.write( "B. x^2 - 2x +1 = 0\r<BR>\n" );
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document.write( "C. x^2 - 4x + 4 = 0\r<BR>\n" );
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document.write( "D. x^2 - 5x + 6 = 0\r<BR>\n" );
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document.write( "E. x^2 + 4x + 4 = 0\r<BR>\n" );
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document.write( "This is a question from a Practice Subject Test I'm taking.  The roots of what? How would you compare the sum of the said roots and the product of them?  This confuses me.\r<BR>\n" );
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document.write( "Thank you very much for any help you can give me! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #387691 by </B><A HREF=/tutors/aboutme.mpl?userid=MathTherapy><B>MathTherapy(10552)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathTherapy><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=MathTherapy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=387691\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For which of the following equations is it true that the sum of the roots equals the product of the roots?\r<BR>\n" );
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document.write( "A. x^2 - 4 = 0\r<BR>\n" );
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document.write( "B. x^2 - 2x +1 = 0\r<BR>\n" );
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document.write( "C. x^2 - 4x + 4 = 0\r<BR>\n" );
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document.write( "D. x^2 - 5x + 6 = 0\r<BR>\n" );
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document.write( "E. x^2 + 4x + 4 = 0\r<BR>\n" );
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document.write( "This is a question from a Practice Subject Test I'm taking.  The roots of what? How would you compare the sum of the said roots and the product of them?  This confuses me.\r<BR>\n" );
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document.write( "Thank you very much for any help you can give me!\r<BR>\n" );
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document.write( "A quadratic equation (one of the above) is in the form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The sum of the roots (solution for x) is determined by the formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"S+=+%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the product is determined by the formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P+=+c%2Fa ALIGN=MIDDLE  ALT=\"P+=+c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the 1st quadratic, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = 1; b = 0; and c = - 4\r<BR>\n" );
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document.write( "Therefore, for this quadratic, the sum of the roots becomes: S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+0%29%2F1%29 ALIGN=MIDDLE  ALT=\"%28-+0%29%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 0\r<BR>\n" );
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document.write( "The product of this quadratic's roots = P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+4%29%2F1 ALIGN=MIDDLE  ALT=\"%28-+4%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or - 4\r<BR>\n" );
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document.write( "The sum (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280%29 ALIGN=MIDDLE  ALT=\"highlight%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of this quadratic's roots is NOT EQUAL to its roots' product (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-+4%29 ALIGN=MIDDLE  ALT=\"highlight%28-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "B. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the 2nd quadratic, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+%2B+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = 1; b = - 2; and c = 1\r<BR>\n" );
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document.write( "Therefore, for this quadratic, the sum of the roots becomes: S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+-+2%29%2F1%29 ALIGN=MIDDLE  ALT=\"%28-+-+2%29%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 2\r<BR>\n" );
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document.write( "The product of this quadratic's roots = P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2F1 ALIGN=MIDDLE  ALT=\"%281%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 1\r<BR>\n" );
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document.write( "The sum (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282%29 ALIGN=MIDDLE  ALT=\"highlight%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of this quadratic's roots is NOT EQUAL to its roots' product (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%29 ALIGN=MIDDLE  ALT=\"highlight%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "C. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4x+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4x+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the 3rd quadratic, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4x+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4x+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = 1; b = - 4; and c = 4\r<BR>\n" );
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document.write( "Therefore, for this quadratic, the sum of the roots becomes: S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+-+4%29%2F1 ALIGN=MIDDLE  ALT=\"%28-+-+4%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 4\r<BR>\n" );
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document.write( "The product of this quadratic's roots = P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%2F1 ALIGN=MIDDLE  ALT=\"%284%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 4\r<BR>\n" );
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document.write( "The sum (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%284%29 ALIGN=MIDDLE  ALT=\"highlight_green%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of this quadratic's roots is EQUAL to its roots' product (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%284%29 ALIGN=MIDDLE  ALT=\"highlight_green%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "We have found the quadratic equation that has its roots' sum equal to its roots' product. We do not have to proceed, but I will do so just to make sure there are no others.\r<BR>\n" );
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document.write( "D. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+5x+%2B+6+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+5x+%2B+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the 4th quadratic, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+5x+%2B+6+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+5x+%2B+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = 1; b = - 5; and c = 6\r<BR>\n" );
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document.write( "Therefore, for this quadratic, the sum of the roots becomes: S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+-+5%29%2F1 ALIGN=MIDDLE  ALT=\"%28-+-+5%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 5\r<BR>\n" );
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document.write( "The product of this quadratic's roots = P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%29%2F1 ALIGN=MIDDLE  ALT=\"%286%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 6\r<BR>\n" );
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document.write( "The sum (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285%29 ALIGN=MIDDLE  ALT=\"highlight%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of this quadratic's roots is NOT EQUAL to its roots' product (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%286%29 ALIGN=MIDDLE  ALT=\"highlight%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "E. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+4x+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+4x+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the 5th quadratic, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+4x+%2B+4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+4x+%2B+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a = 1; b = 4; and c = 4\r<BR>\n" );
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document.write( "Therefore, for this quadratic, the sum of the roots becomes: S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+b%29%2Fa ALIGN=MIDDLE  ALT=\"%28-+b%29%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or S = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+4%29%2F1 ALIGN=MIDDLE  ALT=\"%28-+4%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or - 4\r<BR>\n" );
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document.write( "The product of this quadratic's roots = P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fa ALIGN=MIDDLE  ALT=\"c%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%29%2F1 ALIGN=MIDDLE  ALT=\"%284%29%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or 4\r<BR>\n" );
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document.write( "The sum (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28-+4%29 ALIGN=MIDDLE  ALT=\"highlight%28-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) of this quadratic's roots is NOT EQUAL to its roots' product (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%284%29 ALIGN=MIDDLE  ALT=\"highlight%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "We could've chosen to solve the quadratics and determine their roots, after which we'd add and multiply their individual roots to see if they're equal, but this would've been too time-consuming, so this is an easier method.\r<BR>\n" );
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document.write( "Send comments and “thank-yous” to “D” at  MathMadEzy@aol.com </SPAN>\n" );
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