document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1124033>Question 1124033</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find k so that (k + 1)x^2 + 5x + 2k - 1 = 0 has two real solutions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #740416 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=740416\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <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=+real ALIGN=MIDDLE  ALT=\"+real\"   DISABLED_style= \"max-width:90%; height:auto;\" > solutions.\r<BR>\n" );
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document.write( "use  discriminant:\r<BR>\n" );
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document.write( "if  discriminant is negative,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, our equation has two complex solutions\r<BR>\n" );
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document.write( " if discriminant is positive,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+b%5E2+-+4ac%3E+0 ALIGN=MIDDLE  ALT=\"+b%5E2+-+4ac%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , our equation has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=two ALIGN=MIDDLE  ALT=\"two\"   DISABLED_style= \"max-width:90%; height:auto;\" > solutions which they are real \r<BR>\n" );
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document.write( " if discriminant is zero, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac=+0 ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , our equation has one solution which is real \r<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...here, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%28k+%2B+1%29 ALIGN=MIDDLE  ALT=\"a=%28k+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2k+-+1+ ALIGN=MIDDLE  ALT=\"c=2k+-+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E2+-+4%2A%28k%2B1%29%282k+-+1%29%3E0 ALIGN=MIDDLE  ALT=\"5%5E2+-+4%2A%28k%2B1%29%282k+-+1%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25+-+%284k%2B4%29%282k+-+1%29%3E0 ALIGN=MIDDLE  ALT=\"25+-+%284k%2B4%29%282k+-+1%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25+-+%288k%5E2+-+4k%2B8k-4%29%3E0 ALIGN=MIDDLE  ALT=\"25+-+%288k%5E2+-+4k%2B8k-4%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25=8k%5E2+%2B4k-4 ALIGN=MIDDLE  ALT=\"25=8k%5E2+%2B4k-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8k%5E2+%2B4k-4-25%3E0 ALIGN=MIDDLE  ALT=\"8k%5E2+%2B4k-4-25%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8k%5E2+%2B4k-29%3E0 ALIGN=MIDDLE  ALT=\"8k%5E2+%2B4k-29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "use quadratic formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-b%2B-sqrt%28b%5E2+-+4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"k=%28-b%2B-sqrt%28b%5E2+-+4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4%2B-sqrt%284%5E2+-+4%2A8%28-29%29%29%29%2F%282%2A8%29 ALIGN=MIDDLE  ALT=\"k=%28-4%2B-sqrt%284%5E2+-+4%2A8%28-29%29%29%29%2F%282%2A8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4%2B-sqrt%2816+%2B32%2A29%29%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-4%2B-sqrt%2816+%2B32%2A29%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4%2B-sqrt%28944%29%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-4%2B-sqrt%28944%29%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4%2B-30.72%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-4%2B-30.72%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4%2B30.72%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-4%2B30.72%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=26.72%2F16 ALIGN=MIDDLE  ALT=\"k=26.72%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1.67 ALIGN=MIDDLE  ALT=\"k=1.67\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-4-30.72%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-4-30.72%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=%28-34.72%29%2F16 ALIGN=MIDDLE  ALT=\"k=%28-34.72%29%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2.17 ALIGN=MIDDLE  ALT=\"k=-2.17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "so, we will have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2+-+4ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > if  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2.17 ALIGN=MIDDLE  ALT=\"k=-2.17\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1.67 ALIGN=MIDDLE  ALT=\"k=1.67\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "\n" );
document.write( "check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2.17 ALIGN=MIDDLE  ALT=\"k=-2.17\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28-2.17+%2B+1%29x%5E2+%2B+5x+%2B+2%28-2.17%29+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%28-2.17+%2B+1%29x%5E2+%2B+5x+%2B+2%28-2.17%29+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1.17x%5E2+%2B+5x+-+4.34+-+1+=+0 ALIGN=MIDDLE  ALT=\"+-1.17x%5E2+%2B+5x+-+4.34+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1.17x%5E2+%2B+5x+-+5.34++=+0 ALIGN=MIDDLE  ALT=\"+-1.17x%5E2+%2B+5x+-+5.34++=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "solutions are:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >&#8776;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.10 ALIGN=MIDDLE  ALT=\"2.10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >&#8776;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.18 ALIGN=MIDDLE  ALT=\"2.18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "graph:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C600%2C+-1%2C3%2C+-1%2C+1%2C%0D%0Acircle%282.1%2C0%2C.03%29%2Ccircle%282.18%2C0%2C.03%29%2C%0D%0A+graph%28+600%2C600%2C+-1%2C3%2C+-1%2C+1%2C+-1.17x%5E2+%2B+5x+-+5.34%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%28k+%2B+1%29x%5E2+%2B+5x+%2B+2k+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1.67 ALIGN=MIDDLE  ALT=\"k=1.67\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281.67+%2B+1%29x%5E2+%2B+5x+%2B+2%281.67%29+-+1+=+0 ALIGN=MIDDLE  ALT=\"+%281.67+%2B+1%29x%5E2+%2B+5x+%2B+2%281.67%29+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2.67x%5E2+%2B+5x+%2B+3.34+-+1+=+0 ALIGN=MIDDLE  ALT=\"+2.67x%5E2+%2B+5x+%2B+3.34+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2.67x%5E2+%2B+5x+%2B+2.34++=+0 ALIGN=MIDDLE  ALT=\"+2.67x%5E2+%2B+5x+%2B+2.34++=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "solutions:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >&#8776;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.95 ALIGN=MIDDLE  ALT=\"-0.95\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >&#8776;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.92 ALIGN=MIDDLE  ALT=\"-0.92\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C600%2C+-2%2C1%2C+-1%2C+1%2C%0D%0Acircle%28-0.95%2C0%2C.02%29%2Ccircle%28-0.92%2C0%2C.02%29%2C%0D%0A+graph%28+600%2C600%2C+-2%2C1%2C+-1%2C+1%2C+2.67x%5E2+%2B+5x+%2B+2.34%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );