document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83213>Question 83213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the center of the Conic<BR>\n" );
document.write( "3x^2+4y^2+18x-32y-5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60037 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=60037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume you mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=0 ALIGN=MIDDLE  ALT=\"3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=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=3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=0 ALIGN=MIDDLE  ALT=\"3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+-+5++=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given conic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+=+5+ ALIGN=MIDDLE  ALT=\"3+%2A+x+%5E+2+%2B+4+%2A+y+%5E+2+%2B+1+8+%2A+x+-+3+2+%2A+y+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add 5 to both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+3+%2A+x+%5E+2+%2B+1+8+%2A+x+%29+%2B+%28+4+%2A+y+%5E+2+-+3+2+%2A+y+%29+=+5+ ALIGN=MIDDLE  ALT=\"%28+3+%2A+x+%5E+2+%2B+1+8+%2A+x+%29+%2B+%28+4+%2A+y+%5E+2+-+3+2+%2A+y+%29+=+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%5E+2+%2B+6+%2A+x+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%29+=+-+5+ ALIGN=MIDDLE  ALT=\"3+%28+x+%5E+2+%2B+6+%2A+x+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%29+=+-+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor a 3 out of the first parenthesis<BR>\n" );
document.write( "and factor a 4 out of the second parenthesis.\r<BR>\n" );
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document.write( "Now we must complete the individual squares inside the parenthesis:\r<BR>\n" );
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document.write( " For instance to complete <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%5E+2+%2B+6+%2A+x+%29 ALIGN=MIDDLE  ALT=\"3+%28+x+%5E+2+%2B+6+%2A+x+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, take half of 6 and square it (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2F2%29%5E2=%283%29%5E2=9 ALIGN=MIDDLE  ALT=\"%286%2F2%29%5E2=%283%29%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 9. Now add 9 inside the parenthesis like this: \r<BR>\n" );
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document.write( "Since you really added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A9 ALIGN=MIDDLE  ALT=\"3%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the entire left side, we must add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A9 ALIGN=MIDDLE  ALT=\"3%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the right side also. Now lets complete  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%28+y+%5E+2+-+8+%2A+y+%29+ ALIGN=MIDDLE  ALT=\"4+%28+y+%5E+2+-+8+%2A+y+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to complete <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%28+y+%5E+2+-+8+%2A+y+%29+ ALIGN=MIDDLE  ALT=\"4+%28+y+%5E+2+-+8+%2A+y+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, take half of -8 and square it (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8%2F2%29%5E2=%28-4%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28-8%2F2%29%5E2=%28-4%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 16. Now add 16 inside the parenthesis like this: \r<BR>\n" );
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document.write( "Since you really added <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A16 ALIGN=MIDDLE  ALT=\"4%2A16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the entire left side, we must add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A16 ALIGN=MIDDLE  ALT=\"4%2A16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the right side also.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%5E+2+%2B+6+%2A+x+%2B+9+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%2B+1+6+%29+=+-+5+%2B+9+%2A+3+%2B+1+6+%2A+4+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%5E+2+%2B+6+%2A+x+%2B+9+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%2B+1+6+%29+=+8+6+ ALIGN=MIDDLE  ALT=\"3+%28+x+%5E+2+%2B+6+%2A+x+%2B+9+%29+%2B+4+%28+y+%5E+2+-+8+%2A+y+%2B+1+6+%29+=+8+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+=+8+6+ ALIGN=MIDDLE  ALT=\"3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+=+8+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the individual groups on the left side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+%29+%2F+8+6+=+8+6+%2F+8+6+ ALIGN=MIDDLE  ALT=\"%28+3+%28+x+%2B+3+%29+%5E+2+%2B+4+%28+y+-+4+%29+%5E+2+%29+%2F+8+6+=+8+6+%2F+8+6+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 86\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+ ALIGN=MIDDLE  ALT=\"3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction into standard form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-%24k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-%24k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the equation is an ellipse where the center is (-3,4) \r<BR>\n" );
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document.write( "Here's the graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing+%28500%2C500%2C-10%2C+10%2C+-10%2C+10%2C%0D%0A+++ellipse%28-3%2C4%2C2%2A5.35412613473634%2C2%2A4.63680924774785%29%2C%0D%0A+++circle%28-3%2C4%2C0.08%29%2C%0D%0A+++circle%28-3%2C4%2C0.05%29%2C%0D%0A+++graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C0%29%0D%0A+++%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+ ALIGN=MIDDLE  ALT=\"3+%28+x+%2B+3+%29+%5E+2+%2F+8+6+%2B+4+%28+y+-+4+%29+%5E+2+%2F+8+6+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the center (-3,4)<BR>\n" );
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