document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=10203>Question 10203</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the points of intersection of the graphs.\r<BR>\n" );
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document.write( "x^2 - y^2 - 2x + 2y = 0<BR>\n" );
document.write( "4x^2 + y^2 - 8x = 0\r<BR>\n" );
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document.write( "Please and thank you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #5411 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=5411\">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+-+y%5E2+-+2x+%2B+2y+=+0+ ALIGN=MIDDLE  ALT=\"x%5E2+-+y%5E2+-+2x+%2B+2y+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+y%5E2+-+8x+=+0+ ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+y%5E2+-+8x+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " In fact, the 1st equation is two-lines and the 2nd one is an ellipse<BR>\n" );
document.write( " Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+y%5E2+-+8x+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+y%5E2+-+8x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-1%29%5E2+%2B+y%5E2+=4+ ALIGN=MIDDLE  ALT=\"4%28x-1%29%5E2+%2B+y%5E2+=4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2+%2B+y%5E2%2F4+=+1+ ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2+%2B+y%5E2%2F4+=+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an ellipse centered at (1,0).\r<BR>\n" );
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document.write( " They may have at most 4 points of intersectins.\r<BR>\n" );
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document.write( "[The point is to cancel variable y]<BR>\n" );
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document.write( "Using complete squares to covert the 1st equation to:<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-1%29%5E2+-%28y-1%29%5E2+=+0 ALIGN=MIDDLE  ALT=\"+%28x-1%29%5E2+-%28y-1%29%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...(1)\r<BR>\n" );
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document.write( " Keep the 2nd equation as before  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+y%5E2+-+8x+=+0+ ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+y%5E2+-+8x+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...(2)<BR>\n" );
document.write( " From(1),we have y-1 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%2B-+%28x-1%29 ALIGN=MIDDLE  ALT=\"0+%2B-+%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( +/-means postive or negative)<BR>\n" );
document.write( " When y = x-1 + 1 = x, goto (2), we have<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+-+8x+=+0+ ALIGN=MIDDLE  ALT=\"5x%5E2+-+8x+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,hence x = 0 or 8/5 (why??)\r<BR>\n" );
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document.write( " Then,we get y = 0 or 8/5 (since y = x)\r<BR>\n" );
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document.write( " When y = -x+1 + 1 = -x+2, goto (2), we have<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+%28x-2%29%5E2+-8x+=+0+ ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+%28x-2%29%5E2+-8x+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( " Or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+-+12x+%2B+4+=+0+ ALIGN=MIDDLE  ALT=\"5x%5E2+-+12x+%2B+4+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( " Factoring: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x-2%29%28x-2%29=+0+ ALIGN=MIDDLE  ALT=\"%285x-2%29%28x-2%29=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " So,x = 2/5 or 2 <BR>\n" );
document.write( " and then, y = 8/5 or 0.\r<BR>\n" );
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document.write( " Thus,the four points of intersection are<BR>\n" );
document.write( " (0,0),(8/5,8/5), (2/5, 8/5), and (2,0)\r<BR>\n" );
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document.write( " The graphs as:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-1%2C+5%2C+-4%2C+4%2C+x%2C+-x%2B2%2C+2%2Asqrt%282x-x%5E2%29%2C%0D%0A+-2%2Asqrt%282x-x%5E2%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " Kenny\r<BR>\n" );
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