document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=909043>Question 909043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are two positive integers c for which the equation 5x^2+11x+c=0 has rational solutions. What is the product of those two values of c? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #551559 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=551559\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> in order for the solutions to be rational, (b^2 - 4ac) must be a perfect square.\r<BR>\n" );
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document.write( "the equation is 5x^2 + 11x + c = 0\r<BR>\n" );
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document.write( "a = 5<BR>\n" );
document.write( "b = 11<BR>\n" );
document.write( "c = c\r<BR>\n" );
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document.write( "the quadratic formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B+sqrt%28b%5E2-4ac%29%29+%2F+%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B+sqrt%28b%5E2-4ac%29%29+%2F+%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "replace a with 5 and b with 11 and you get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-11+%2B+sqrt%28121-20c%29%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-11+%2B+sqrt%28121-20c%29%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11+-+sqrt%28121-20c%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-11+-+sqrt%28121-20c%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(121-20c) must be a perfect square.\r<BR>\n" );
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document.write( "when c = 0, 121 is a perfect square.<BR>\n" );
document.write( "but c has to be positive and 0 is not positive.\r<BR>\n" );
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document.write( "when c = 1, 121 - 20 = 101 is not a perfect square.<BR>\n" );
document.write( "when c = 2, 121 - 40 = 81 is a perfect square **************<BR>\n" );
document.write( "when c = 3, 121 - 60 = 61 is not a perfect square.<BR>\n" );
document.write( "when c = 4, 121 - 80 = 41 is not a perfect square.<BR>\n" );
document.write( "when c = 5, 121 - 100 = 21 is not a perfect square.<BR>\n" );
document.write( "when c = 6, 121 - 120 = 1 is a perfect square **************\r<BR>\n" );
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document.write( "looks like your solution is c = 2 and c = 6\r<BR>\n" );
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document.write( "when c = 2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-11+%2B+sqrt%28121-20c%29%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-11+%2B+sqrt%28121-20c%29%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11+-+sqrt%28121-20c%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-11+-+sqrt%28121-20c%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-11+%2B+sqrt%28121-40%29%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-11+%2B+sqrt%28121-40%29%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11+-+sqrt%28121-40%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-11+-+sqrt%28121-40%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-11+%2B+sqrt%2881%29%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-11+%2B+sqrt%2881%29%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11+-+sqrt%2881%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-11+-+sqrt%2881%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-11+%2B+9%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-11+%2B+9%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-11+-+9%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-11+-+9%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-2%29+%2F+%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-2%29+%2F+%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-20%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"%28-20%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-1%2F5 ALIGN=MIDDLE  ALT=\"x+=+-1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "you can confirm the solution is good by replacing c with 2 in the equation and then solving for f(-2/10) and f(-2)\r<BR>\n" );
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document.write( "the equation you started with is:\r<BR>\n" );
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document.write( "5x^2 + 11x + 2 = 0\r<BR>\n" );
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document.write( "set f(x) = 5x^2 + 11x + 2\r<BR>\n" );
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document.write( "f(-2/10) = 0\r<BR>\n" );
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document.write( "f(-2) = 0\r<BR>\n" );
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document.write( "I confirmed with my calculator and I also graphed the equation to show the solution graphically.\r<BR>\n" );
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document.write( "the graph when c = 2 is shown below.<BR>\n" );
document.write( "the zero points are at x = -2 and x = -1/5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/100303.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the graph when c = 6 is shown below.<BR>\n" );
document.write( "the zero points are at x = -6/5 and x = -1.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/100304.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
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