document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=927857>Question 927857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm struggling with this question on a past paper:\r<BR>\n" );
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document.write( "A curve has equation y = 2x^2 - x - 1 and a line has equation y = k(2x -3) , where<BR>\n" );
document.write( "k is a constant.\r<BR>\n" );
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document.write( "The x-coordinate of any point of intersection of the curve and the line satisfies the equation<BR>\n" );
document.write( "2x^2 - (2k +1)x + 3k - 1 = 0 \r<BR>\n" );
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document.write( "The curve and the line intersect at two distinct points. <BR>\n" );
document.write( "Show that 4k^2 - 20k + 9 > 0.\r<BR>\n" );
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document.write( "If someone would be able to explain the steps needed that would be great, as I'm drawing a complete blank on this.\r<BR>\n" );
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document.write( "Many thanks, <BR>\n" );
document.write( "R </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #563296 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=563296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are told that \"The curve and the line intersect at two distinct points.\" so that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-%282k+%2B1%29x+%2B+3k-1+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-%282k+%2B1%29x+%2B+3k-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has two distinct roots/solutions for x.\r<BR>\n" );
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document.write( "That only happens when the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater than zero, ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%3E0 ALIGN=MIDDLE  ALT=\"D%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In general, the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+b%5E2+-+4ac ALIGN=MIDDLE  ALT=\"D+=+b%5E2+-+4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-%282k+%2B1%29x+%2B+3k-1+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-%282k+%2B1%29x+%2B+3k-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2 ALIGN=MIDDLE  ALT=\"a+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-%282k%2B1%29 ALIGN=MIDDLE  ALT=\"b+=+-%282k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+3k-1 ALIGN=MIDDLE  ALT=\"c+=+3k-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug in those values of a,b,c and simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+b%5E2+-+4ac ALIGN=MIDDLE  ALT=\"D+=+b%5E2+-+4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+%28-%282k%2B1%29%29%5E2+-+4%2A2%283k-1%29 ALIGN=MIDDLE  ALT=\"D+=+%28-%282k%2B1%29%29%5E2+-+4%2A2%283k-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+%282k%2B1%29%5E2+-+24k+%2B+8 ALIGN=MIDDLE  ALT=\"D+=+%282k%2B1%29%5E2+-+24k+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+4k%5E2%2B4k%2B1+-+24k+%2B+8 ALIGN=MIDDLE  ALT=\"D+=+4k%5E2%2B4k%2B1+-+24k+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+4k%5E2-20k%2B9 ALIGN=MIDDLE  ALT=\"D+=+4k%5E2-20k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%3E0 ALIGN=MIDDLE  ALT=\"D%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+4k%5E2-20k%2B9 ALIGN=MIDDLE  ALT=\"D+=+4k%5E2-20k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we know <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2-20k%2B9%3E0 ALIGN=MIDDLE  ALT=\"4k%5E2-20k%2B9%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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