document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132645>Question 132645</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two lines have slopes 2k-4 and k+6. What value(s) of k will produce perpendicular lines?  I think the answer is k=-4+or- sqrt66/2???????? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96937 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=96937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Their slopes will be perpendicular when their product is -1. So we have the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k-4%29%2A%28k%2B6%29=-1 ALIGN=MIDDLE  ALT=\"%282k-4%29%2A%28k%2B6%29=-1\"   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=2k%5E2%2B8k-24=-1 ALIGN=MIDDLE  ALT=\"2k%5E2%2B8k-24=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2k%5E2%2B8k-23=0 ALIGN=MIDDLE  ALT=\"2k%5E2%2B8k-23=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 1 to both sides\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for k:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ak%5E2%2Bbk%2Bc=0 ALIGN=MIDDLE  ALT=\"ak%5E2%2Bbk%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\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%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"k+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ak%5E2%2B8%2Ak-23=0 ALIGN=MIDDLE  ALT=\"2%2Ak%5E2%2B8%2Ak-23=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <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=8 ALIGN=MIDDLE  ALT=\"b=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-23 ALIGN=MIDDLE  ALT=\"c=-23\"   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=k+=+%28-8+%2B-+sqrt%28+%288%29%5E2-4%2A2%2A-23+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+sqrt%28+%288%29%5E2-4%2A2%2A-23+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=2, b=8, and c=-23\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+%2B-+sqrt%28+64-4%2A2%2A-23+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+sqrt%28+64-4%2A2%2A-23+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 8 to get 64  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+%2B-+sqrt%28+64%2B184+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+sqrt%28+64%2B184+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-23%2A2 ALIGN=MIDDLE  ALT=\"-4%2A-23%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=184 ALIGN=MIDDLE  ALT=\"184\"   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=k+=+%28-8+%2B-+sqrt%28+248+%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+sqrt%28+248+%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F4 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B-+2%2Asqrt%2862%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 2 to get 4\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+%2B+2%2Asqrt%2862%29%29%2F4 ALIGN=MIDDLE  ALT=\"k+=+%28-8+%2B+2%2Asqrt%2862%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-8+-+2%2Asqrt%2862%29%29%2F4 ALIGN=MIDDLE  ALT=\"k+=+%28-8+-+2%2Asqrt%2862%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-8%2F4%2B2%2Asqrt%2862%29%2F4 ALIGN=MIDDLE  ALT=\"k=-8%2F4%2B2%2Asqrt%2862%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-8%2F4-2%2Asqrt%2862%29%2F4 ALIGN=MIDDLE  ALT=\"k=-8%2F4-2%2Asqrt%2862%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2%2Bsqrt%2862%29%2F2 ALIGN=MIDDLE  ALT=\"k=-2%2Bsqrt%2862%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2-sqrt%2862%29%2F2 ALIGN=MIDDLE  ALT=\"k=-2-sqrt%2862%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So these expressions approximate to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1.93700393700591 ALIGN=MIDDLE  ALT=\"k=1.93700393700591\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-5.93700393700591 ALIGN=MIDDLE  ALT=\"k=-5.93700393700591\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solutions are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1.93700393700591 ALIGN=MIDDLE  ALT=\"k=1.93700393700591\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-5.93700393700591 ALIGN=MIDDLE  ALT=\"k=-5.93700393700591\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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