document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184257>Question 1184257</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that the line x + y = q will intersect the curve x^2 - 2x + 2y^2 = 3 in two distinct points if q^2 < 2q + 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814811 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814811\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Show that the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y+=+q ALIGN=MIDDLE  ALT=\"x+%2B+y+=+q\"   DISABLED_style= \"max-width:90%; height:auto;\" > will intersect the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+%2B+2y%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+2y%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > in two distinct points if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%5E2+%3C+2q+%2B+5 ALIGN=MIDDLE  ALT=\"q%5E2+%3C+2q+%2B+5\"   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=+x+%2B+y+=+q+ ALIGN=MIDDLE  ALT=\"+x+%2B+y+=+q+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......eq.1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+%2B+2y%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+2y%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........eq.2<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++y+=+q-x+ ALIGN=MIDDLE  ALT=\"++y+=+q-x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......eq.1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+%2B+2y%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+2y%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........eq.2...substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=x%5E2+-+2x+%2B+2%28q-x+%29%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+2%28q-x+%29%5E2+=+3\"   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=x%5E2+-+2x+%2B+2q%5E2+-+4qx+%2B+2x%5E2=+3 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+%2B+2q%5E2+-+4qx+%2B+2x%5E2=+3\"   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=3x%5E2+-+2x-+4qx+%2B+2q%5E2++-+3=0 ALIGN=MIDDLE  ALT=\"3x%5E2+-+2x-+4qx+%2B+2q%5E2++-+3=0\"   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=3x%5E2+-+%282%2B+4q%29x+%2B+%282q%5E2++-+3%29=0 ALIGN=MIDDLE  ALT=\"3x%5E2+-+%282%2B+4q%29x+%2B+%282q%5E2++-+3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "use discriminant<BR>\n" );
document.write( "the line will intersect  the curve in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=two ALIGN=MIDDLE  ALT=\"two\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+distinct ALIGN=MIDDLE  ALT=\"+distinct\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+points ALIGN=MIDDLE  ALT=\"+points\"   DISABLED_style= \"max-width:90%; height:auto;\" > if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b%5E2-4ac+%3E0%29 ALIGN=MIDDLE  ALT=\"highlight%28b%5E2-4ac+%3E0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "in your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-+%284q%2B2%29 ALIGN=MIDDLE  ALT=\"b=-+%284q%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=%282q%5E2++-+3%29 ALIGN=MIDDLE  ALT=\"c=%282q%5E2++-+3%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=%28-+%284q%2B2%29%29%5E2-4%2A3%282q%5E2++-+3%29+%3E0 ALIGN=MIDDLE  ALT=\"%28-+%284q%2B2%29%29%5E2-4%2A3%282q%5E2++-+3%29+%3E0\"   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=16q%5E2+%2B+16q+%2B+4-24q%5E2+%2B36+%3E0 ALIGN=MIDDLE  ALT=\"16q%5E2+%2B+16q+%2B+4-24q%5E2+%2B36+%3E0\"   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=16q%5E2+%2B+16q+%2B+4-24q%5E2+%2B36+%3E0 ALIGN=MIDDLE  ALT=\"16q%5E2+%2B+16q+%2B+4-24q%5E2+%2B36+%3E0\"   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=-8q%5E2+%2B+16q+%2B+40+%3E0 ALIGN=MIDDLE  ALT=\"-8q%5E2+%2B+16q+%2B+40+%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"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=-q%5E2+%2B+2q+%2B+5+%3E0 ALIGN=MIDDLE  ALT=\"-q%5E2+%2B+2q+%2B+5+%3E0\"   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=++2q+%2B+5+%3Eq%5E2 ALIGN=MIDDLE  ALT=\"++2q+%2B+5+%3Eq%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28q%5E2%3C2q+%2B+5%29 ALIGN=MIDDLE  ALT=\"highlight%28q%5E2%3C2q+%2B+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );