document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=407387>Question 407387</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the quadratic system.\r<BR>\n" );
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document.write( "Solve algebraically: 4x^2-9y^2=108 and xy=-12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287210 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=287210\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> xy=-12\r<BR>\n" );
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document.write( "y=-12/x\r<BR>\n" );
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document.write( "4x^2-9y^2=108\r<BR>\n" );
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document.write( "4x^2-9(-12/x)^2=108\r<BR>\n" );
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document.write( "4x^2-9(144/x^2)=108\r<BR>\n" );
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document.write( "4x^4-1296=108x^2\r<BR>\n" );
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document.write( "4x^4-108x^2-1296=0\r<BR>\n" );
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document.write( "4z^2-108z-1296=0\r<BR>\n" );
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document.write( "Notice we have a quadratic equation in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=az%5E2%2Bbz%2Bc ALIGN=MIDDLE  ALT=\"az%5E2%2Bbz%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-108 ALIGN=MIDDLE  ALT=\"b=-108\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1296 ALIGN=MIDDLE  ALT=\"c=-1296\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for z\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28-%28-108%29+%2B-+sqrt%28+%28-108%29%5E2-4%284%29%28-1296%29+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28-%28-108%29+%2B-+sqrt%28+%28-108%29%5E2-4%284%29%28-1296%29+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-108 ALIGN=MIDDLE  ALT=\"b=-108\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-1296 ALIGN=MIDDLE  ALT=\"c=-1296\"   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=z+=+%28108+%2B-+sqrt%28+%28-108%29%5E2-4%284%29%28-1296%29+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+%28-108%29%5E2-4%284%29%28-1296%29+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-108 ALIGN=MIDDLE  ALT=\"-108\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=108 ALIGN=MIDDLE  ALT=\"108\"   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=z+=+%28108+%2B-+sqrt%28+11664-4%284%29%28-1296%29+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+11664-4%284%29%28-1296%29+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-108 ALIGN=MIDDLE  ALT=\"-108\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11664 ALIGN=MIDDLE  ALT=\"11664\"   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=z+=+%28108+%2B-+sqrt%28+11664--20736+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+11664--20736+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%284%29%28-1296%29 ALIGN=MIDDLE  ALT=\"4%284%29%28-1296%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20736 ALIGN=MIDDLE  ALT=\"-20736\"   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=z+=+%28108+%2B-+sqrt%28+11664%2B20736+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+11664%2B20736+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2811664--20736%29 ALIGN=MIDDLE  ALT=\"sqrt%2811664--20736%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2811664%2B20736%29 ALIGN=MIDDLE  ALT=\"sqrt%2811664%2B20736%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=z+=+%28108+%2B-+sqrt%28+32400+%29%29%2F%282%284%29%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+32400+%29%29%2F%282%284%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11664 ALIGN=MIDDLE  ALT=\"11664\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20736 ALIGN=MIDDLE  ALT=\"20736\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32400 ALIGN=MIDDLE  ALT=\"32400\"   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=z+=+%28108+%2B-+sqrt%28+32400+%29%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+sqrt%28+32400+%29%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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=z+=+%28108+%2B-+180%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B-+180%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32400 ALIGN=MIDDLE  ALT=\"32400\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180 ALIGN=MIDDLE  ALT=\"180\"   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=z+=+%28108+%2B+180%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+%2B+180%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28108+-+180%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=+%28108+-+180%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+%28288%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=+%28288%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=++%28-72%29%2F%288%29 ALIGN=MIDDLE  ALT=\"z+=++%28-72%29%2F%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+36 ALIGN=MIDDLE  ALT=\"z+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-9 ALIGN=MIDDLE  ALT=\"z+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. \r<BR>\n" );
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document.write( "So the answers in terms of z are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+36 ALIGN=MIDDLE  ALT=\"z+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-9 ALIGN=MIDDLE  ALT=\"z+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now above, I let z = x^2. So x^2 = 36 and x^2 = -9\r<BR>\n" );
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document.write( "Solve for x in each equation to get: x = 6, x = -6, x = 3i, or x = -3i\r<BR>\n" );
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document.write( "So the four solutions for x are x = 6, x = -6, x = 3i, or x = -3i\r<BR>\n" );
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document.write( "Now plug all of these x values into y = -12/x to find the corresponding values of y. I'll let you do this.\r<BR>\n" );
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document.write( "In the end, you should have 4 ordered pairs. \r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Jim<BR>\n" );
document.write( " </SPAN>\n" );
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