document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=155037>Question 155037</A>: <I><SPAN STYLE=\"word-break: break-all;\"> algebra of linear and quadratic expressions.\r<BR>\n" );
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document.write( "find the values of k for which the quadratic 2x^2+kx+9=0 has one real solution, and another with no real solutions. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #114144 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=114144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <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;\" > Start with the discriminant formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=k%5E2-4%282%29%289%29 ALIGN=MIDDLE  ALT=\"D=k%5E2-4%282%29%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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=k ALIGN=MIDDLE  ALT=\"b=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   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=k%5E2-8%289%29 ALIGN=MIDDLE  ALT=\"D=k%5E2-8%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 4 and 2 to get 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=k%5E2-72 ALIGN=MIDDLE  ALT=\"D=k%5E2-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 8 and 9 to get 72\r<BR>\n" );
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document.write( "So we'll use the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=k%5E2-72 ALIGN=MIDDLE  ALT=\"D=k%5E2-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve the following:\r<BR>\n" );
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document.write( "1) \"find the values of k for which the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Bkx%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2Bkx%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has one real solution\"\r<BR>\n" );
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document.write( "If a quadratic has one real solution, then the discriminant is equal to zero. So this means that <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=k%5E2-72 ALIGN=MIDDLE  ALT=\"D=k%5E2-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the previous equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=k%5E2-72 ALIGN=MIDDLE  ALT=\"0=k%5E2-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=k%5E2 ALIGN=MIDDLE  ALT=\"72=k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 72 to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B-sqrt%2872%29=k ALIGN=MIDDLE  ALT=\"0%2B-sqrt%2872%29=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=sqrt%2872%29 ALIGN=MIDDLE  ALT=\"k=sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-sqrt%2872%29 ALIGN=MIDDLE  ALT=\"k=-sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the \"plus/minus\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k=2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k=-2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root\r<BR>\n" );
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document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k=2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k=-2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the discriminant is equal to zero. This means that the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B2%2Asqrt%2818%29x%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B2%2Asqrt%2818%29x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-2%2Asqrt%2818%29x%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2-2%2Asqrt%2818%29x%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > only has one real solution.\r<BR>\n" );
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document.write( "2) \"find the values of k for which the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Bkx%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2Bkx%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > no real solutions\"\r<BR>\n" );
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document.write( "If a quadratic has no real solutions, this means that the discriminant is less than zero. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%3C0 ALIGN=MIDDLE  ALT=\"D%3C0\"   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=k%5E2-72 ALIGN=MIDDLE  ALT=\"D=k%5E2-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the previous equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2-72%3C0 ALIGN=MIDDLE  ALT=\"k%5E2-72%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%3C0 ALIGN=MIDDLE  ALT=\"D%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the right side is less than zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2%3C72 ALIGN=MIDDLE  ALT=\"k%5E2%3C72\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 72 to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C0%2B-sqrt%2872%29 ALIGN=MIDDLE  ALT=\"k%3C0%2B-sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3Csqrt%2872%29 ALIGN=MIDDLE  ALT=\"k%3Csqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E-sqrt%2872%29 ALIGN=MIDDLE  ALT=\"k%3E-sqrt%2872%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the \"plus/minus\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3C2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k%3C2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E-2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"k%3E-2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%2818%29%3Ck%3C2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%2818%29%3Ck%3C2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Recombine the two inequalities to form one compound inequality\r<BR>\n" );
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document.write( "So if \"k\" is in between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the discriminant is less than zero. This means that if \"k\" is in between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"-2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%2818%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%2818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Bkx%2B9=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2Bkx%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have no solutions\r<BR>\n" );
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