document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=308967>Question 308967</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What type of graph does this equation represent? \r<BR>\n" );
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document.write( "x^2  + 4y^2 - 2y = 8\r<BR>\n" );
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document.write( "THANKS! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #220908 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=220908\">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=x%5E2++%2B+4y%5E2+-+2y+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4y%5E2+-+2y+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28y%5E2+-+%281%2F2%29y%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y%5E2+-+%281%2F2%29y%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor a 4 from the last two terms 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=x%5E2++%2B+4%28y%5E2+-+%281%2F2%29y%2B1%2F16-1%2F16%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y%5E2+-+%281%2F2%29y%2B1%2F16-1%2F16%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now take half of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Square it to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F16 ALIGN=MIDDLE  ALT=\"1%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Add AND subtract this value inside the parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28%28y%5E2+-+%281%2F2%29y%2B1%2F16%29-1%2F16%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28%28y%5E2+-+%281%2F2%29y%2B1%2F16%29-1%2F16%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group up the first three terms inside the parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28%28y-1%2F4%29%5E2-1%2F16%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28%28y-1%2F4%29%5E2-1%2F16%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor that inner group to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-1%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28y-1%2F4%29%5E2\"   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=x%5E2++%2B+4%28y-1%2F4%29%5E2-4%281%2F16%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y-1%2F4%29%5E2-4%281%2F16%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28y-1%2F4%29%5E2-4%2F16+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y-1%2F4%29%5E2-4%2F16+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28y-1%2F4%29%5E2-1%2F4=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y-1%2F4%29%5E2-1%2F4=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28y-1%2F4%29%5E2=+8%2B1%2F4 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y-1%2F4%29%5E2=+8%2B1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4%28y-1%2F4%29%5E2=+33%2F4 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4%28y-1%2F4%29%5E2=+33%2F4\"   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=4x%5E2++%2B+16%28y-1%2F4%29%5E2=+33 ALIGN=MIDDLE  ALT=\"4x%5E2++%2B+16%28y-1%2F4%29%5E2=+33\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term (outside the parenthesis) by the LCD 4 to clear out the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2++%2B+16%28y-1%2F4%29%5E2%29%2F33=+1 ALIGN=MIDDLE  ALT=\"%284x%5E2++%2B+16%28y-1%2F4%29%5E2%29%2F33=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 33 (to make the right side equal to 1).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%29%2F33++%2B+%2816%28y-1%2F4%29%5E2%29%2F33=+1 ALIGN=MIDDLE  ALT=\"%284x%5E2%29%2F33++%2B+%2816%28y-1%2F4%29%5E2%29%2F33=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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=%28x%5E2%29%2F%2833%2F4%29++%2B+%28%28y-1%2F4%29%5E2%29%2F%2833%2F16%29=+1 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%2F%2833%2F4%29++%2B+%28%28y-1%2F4%29%5E2%29%2F%2833%2F16%29=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms in the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%2F%28%28sqrt%2833%29%2F2%29%5E2%29++%2B+%28%28y-1%2F4%29%5E2%29%2F%28%28sqrt%2833%29%2F4%29%5E2%29=+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33%2F4 ALIGN=MIDDLE  ALT=\"33%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2833%29%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%2833%29%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33%2F16 ALIGN=MIDDLE  ALT=\"33%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%2833%29%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%2833%29%2F4%29%5E2\"   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=%28x-0%29%5E2%2F%28%28sqrt%2833%29%2F2%29%5E2%29++%2B+%28%28y-1%2F4%29%5E2%29%2F%28%28sqrt%2833%29%2F4%29%5E2%29=+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Finally, write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-0%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-0%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It might be hard to notice, but the last equation above is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is an ellipse.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-0%29%5E2%2F%28%28sqrt%2833%29%2F2%29%5E2%29++%2B+%28%28y-1%2F4%29%5E2%29%2F%28%28sqrt%2833%29%2F4%29%5E2%29=+1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > is an ellipse\r<BR>\n" );
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document.write( "This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4y%5E2+-+2y+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4y%5E2+-+2y+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also an ellipse (since the two equations are equivalent)\r<BR>\n" );
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document.write( "This is a lot of work to determine the conic. An alternative is to use the following rule:\r<BR>\n" );
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document.write( "For the general conic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0 ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3C0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is an ellipse<BR>\n" );
document.write( "Furthermore, if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=C ALIGN=MIDDLE  ALT=\"A=C\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the conic is also a circle\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC=0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is a parabola\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3E0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is a hyperbola\r<BR>\n" );
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document.write( "In our case, we have the conic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2++%2B+4y%5E2+-+2y+=+8 ALIGN=MIDDLE  ALT=\"x%5E2++%2B+4y%5E2+-+2y+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=1 ALIGN=MIDDLE  ALT=\"A=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=4 ALIGN=MIDDLE  ALT=\"C=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E=-2 ALIGN=MIDDLE  ALT=\"E=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=-8 ALIGN=MIDDLE  ALT=\"F=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (subtract this from both sides). Plug these values into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC ALIGN=MIDDLE  ALT=\"B%5E2-4AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5E2-4%281%29%284%29=-16 ALIGN=MIDDLE  ALT=\"0%5E2-4%281%29%284%29=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is indeed less than 0 meaning that it is an ellipse.\r<BR>\n" );
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document.write( "Personally, I recommend doing it the long way for a while at first so you get a feel of what you are doing. Once you understand what's going on, take the shortcut. </SPAN>\n" );
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