document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206552>Question 1206552</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What value of K will cause <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Kn%5E2+-+3n+%2B+2Kn+%2B+1 ALIGN=MIDDLE  ALT=\"Kn%5E2+-+3n+%2B+2Kn+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to have two zeroes / roots? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844100 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3835)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=844100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Discriminant formula<BR>\n" );
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( "Rule: If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+%3E+0 ALIGN=MIDDLE  ALT=\"d+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the quadratic will have two distinct roots.\r<BR>\n" );
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document.write( "In this case,<BR>\n" );
document.write( "a = k<BR>\n" );
document.write( "b = -3+2k<BR>\n" );
document.write( "c = 1\r<BR>\n" );
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document.write( "It might help to think of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kn%5E2+-+3n+%2B+2kn+%2B+1 ALIGN=MIDDLE  ALT=\"kn%5E2+-+3n+%2B+2kn+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kn%5E2+%2B+%28-3%2B2k%29n+%2B+1 ALIGN=MIDDLE  ALT=\"kn%5E2+%2B+%28-3%2B2k%29n+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so you can match those terms up to the template <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=an%5E2+%2B+bn+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"an%5E2+%2B+bn+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's find the expression for the discriminant.<BR>\n" );
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+=+%28-3%2B2k%29%5E2+-+4k%281%29 ALIGN=MIDDLE  ALT=\"d+=+%28-3%2B2k%29%5E2+-+4k%281%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+=+%289-12k%2B4k%5E2%29+-+4k ALIGN=MIDDLE  ALT=\"d+=+%289-12k%2B4k%5E2%29+-+4k\"   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%5E2-16k%2B9 ALIGN=MIDDLE  ALT=\"d+=+4k%5E2-16k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now consider the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2+-+16k+%2B+9+=+0 ALIGN=MIDDLE  ALT=\"4k%5E2+-+16k+%2B+9+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use the quadratic formula, which I'll leave the scratch work for the student to do, to find the following roots <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%284-sqrt%287%29%29%2F2+=+0.677124 ALIGN=MIDDLE  ALT=\"k+=+%284-sqrt%287%29%29%2F2+=+0.677124\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%284%2Bsqrt%287%29%29%2F2+=+3.322876 ALIGN=MIDDLE  ALT=\"k+=+%284%2Bsqrt%287%29%29%2F2+=+3.322876\"   DISABLED_style= \"max-width:90%; height:auto;\" > Both of those decimal values are approximate.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+%3C+%284-sqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+%3C+%284-sqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+%3E+%284%2Bsqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+%3E+%284%2Bsqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2+-+16k+%2B+9+%3E+0 ALIGN=MIDDLE  ALT=\"4k%5E2+-+16k+%2B+9+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to lead back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+%3E+0 ALIGN=MIDDLE  ALT=\"d+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cause <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kn%5E2+-+3n+%2B+2kn+%2B+1 ALIGN=MIDDLE  ALT=\"kn%5E2+-+3n+%2B+2kn+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to have two distinct roots for variable n.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%284-sqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+%284-sqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%284%2Bsqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+%284%2Bsqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+0 ALIGN=MIDDLE  ALT=\"d+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and it will make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kn%5E2+-+3n+%2B+2kn+%2B+1 ALIGN=MIDDLE  ALT=\"kn%5E2+-+3n+%2B+2kn+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > have exactly one real root for variable n.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-sqrt%287%29%29%2F2+%3C+k+%3C+%284%2Bsqrt%287%29%29%2F2 ALIGN=MIDDLE  ALT=\"%284-sqrt%287%29%29%2F2+%3C+k+%3C+%284%2Bsqrt%287%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then it leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+%3C+0 ALIGN=MIDDLE  ALT=\"d+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and causes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kn%5E2+-+3n+%2B+2kn+%2B+1 ALIGN=MIDDLE  ALT=\"kn%5E2+-+3n+%2B+2kn+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to have no real roots. The two roots would instead be non-real complex numbers.<BR>\n" );
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