document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=892918>Question 892918</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Completing the square <BR>\n" );
document.write( "1. X^4+64y^4<BR>\n" );
document.write( "2. X^4+y^2+25<BR>\n" );
document.write( "3. X^4-11x^2y^2+y^4<BR>\n" );
document.write( " <BR>\n" );
document.write( "Please do help me,\r<BR>\n" );
document.write( "\n" );
document.write( "My teacher's solution goes like this, and I didn't even understand cuz he's too fast.\r<BR>\n" );
document.write( "\n" );
document.write( "Ex. x^4+2x^2y^2+9y^4<BR>\n" );
document.write( "x^4+9y^4+2x^2y^2<BR>\n" );
document.write( "x^4+2x^2(3y^2)+9y^4+2x^2y^2-@x^2(3y^2)<BR>\n" );
document.write( "X^4+6x^2y^2+9y^4+2x^2y^2-6x^2y^2<BR>\n" );
document.write( "(x^2+3y^2)^2-4x^2y^2<BR>\n" );
document.write( "(x^2+3y^2)^2-(2xy)^2<BR>\n" );
document.write( "(x^2+3y^2+2xy)(x^2+3y^2-2xy)\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #541040 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=541040\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Look for terms that are squares. It is not easy to see them at first. It takes practice.<BR>\n" );
document.write( " <BR>\n" );
document.write( "I will over-explain your teacher's solution:<BR>\n" );
document.write( "In <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B2x%5E2y%5E2%2B9y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2B2x%5E2y%5E2%2B9y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > your teacher sees two squares: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%5E2=red%28x%5E4%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%5E2=red%28x%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y2%29%5E2=red%289y%5E4%29 ALIGN=MIDDLE  ALT=\"%283y2%29%5E2=red%289y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The expressions <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y%5E2 ALIGN=MIDDLE  ALT=\"3y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , squared, appear in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29 ALIGN=MIDDLE  ALT=\"red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we add those two expressions and square the sum we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2=%28x%5E2%29%5E2%2B2%28x%5E2%29%283y2%29%2B%283y2%29%5E2=red%28x%5E4%29%2Bgreen%282x%5E2%283y%5E2%29%29%2Bred%289y%5E4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To make the square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2Bgreen%282x%5E2%283y%5E2%29%29%2Bred%289y%5E4%29 ALIGN=MIDDLE  ALT=\"red%28x%5E4%29%2Bgreen%282x%5E2%283y%5E2%29%29%2Bred%289y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > appear,<BR>\n" );
document.write( "your teacher does some plastic surgery to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29 ALIGN=MIDDLE  ALT=\"red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "He takes something out of one end and adds it to another place, just like a plastic surgeon would. <BR>\n" );
document.write( "He adds <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%5E2%283y%5E2%29%29 ALIGN=MIDDLE  ALT=\"green%282x%5E2%283y%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to \"complete the square\",<BR>\n" );
document.write( "and subtracts the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%5E2%283y%5E2%29%29 ALIGN=MIDDLE  ALT=\"green%282x%5E2%283y%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so as not to really change anything.<BR>\n" );
document.write( "Doing that to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29 ALIGN=MIDDLE  ALT=\"red%28x%5E4%29%2B2x%5E2y%5E2%2Bred%289y%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , he gets<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2Bgreen%282x%5E2%283y%5E2%29%29%2Bred%289y%5E4%29%2B2x%5E2y%5E2-green%282x%5E2%283y%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Of course, he knows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282x%5E2%283y%5E2%29%29=green%286x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"green%282x%5E2%283y%5E2%29%29=green%286x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and thinks that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%286x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"green%286x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a more elegant way to write it so he writes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E4%29%2Bgreen%286x%5E2y%5E2%29%2Bred%289y%5E4%29%2B2x%5E2y%5E2-green%286x%5E2y%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The first three terms are his completed square,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2=red%28x%5E4%29%2Bgreen%282x%5E2%283y%5E2%29%29%2Bred%289y%5E4%29=red%28x%5E4%29%2Bgreen%286x%5E2y%5E2%29%2Bred%289y%5E4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and now he writes that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3y%5E2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for short,<BR>\n" );
document.write( "and he should have written<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2%2B2x%5E2y%5E2-green%286x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3y%5E2%29%5E2%2B2x%5E2y%5E2-green%286x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Collecting like terms, he gets<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2-4x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3y%5E2%29%5E2-4x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now he realizes that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2y%5E2=%282xy%29%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2y%5E2=%282xy%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also a square, and re-writes his expression as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2-%282xy%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3y%5E2%29%5E2-%282xy%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "He likes that because now he has a difference of squares,<BR>\n" );
document.write( "and he knows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2-B%5E2=%28A%2BB%29%28A-B%29 ALIGN=MIDDLE  ALT=\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any two expressions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=x%5E2%2B3y%5E2 ALIGN=MIDDLE  ALT=\"A=x%5E2%2B3y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=2xy ALIGN=MIDDLE  ALT=\"B=2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%29%5E2-%282xy%29%5E2=%28%28x%5E2%2B3y%5E2%29%2B%282xy%29%29%28%28x%5E2%2B3y%5E2%29-%282xy%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Of course, we do not need those parentheses around <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B3y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and around <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xy ALIGN=MIDDLE  ALT=\"2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "I just wrote those parentheses so you would see the separate expressions.<BR>\n" );
document.write( "So your teacher writes, without unnecessary parentheses,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B3y%5E2%2B2xy%29%28x%5E2%2B3y%5E2-2xy%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B3y%5E2%2B2xy%29%28x%5E2%2B3y%5E2-2xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B64y%5E4=%28x%5E2%29%5E2%2B%288y%5E2%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E4%2B64y%5E4=%28x%5E2%29%5E2%2B%288y%5E2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (I found two squares that are added).<BR>\n" );
document.write( "That could be part of the square of a sum:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B8y%5E2%29%5E2=%28x%5E2%29%5E2%2Bgreen%282%28x%5E2%29%288y%5E2%29%29%2B%288y%5E2%29%5E2=x%5E4%2Bgreen%2816x%5E2y%5E2%29%2B64y%5E4 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To make the complete square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B16y%5E2%29%5E2=x%5E4%2Bgreen%2816x%5E2y%5E2%29%2B64y%5E4 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B16y%5E2%29%5E2=x%5E4%2Bgreen%2816x%5E2y%5E2%29%2B64y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "appear in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B64y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2B64y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , I add and subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%2816x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"green%2816x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bgreen%2832x%5E2y%5E2%29%2B64y%5E4-green%2832x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E4%2Bgreen%2832x%5E2y%5E2%29%2B64y%5E4-green%2832x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I can re-write that as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B8y%5E2%29%5E2-green%2816x%5E2y%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B8y%5E2%29%5E2-green%2816x%5E2y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284xy%29%5E2 ALIGN=MIDDLE  ALT=\"%284xy%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a also a square, I have the difference of squares<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B8y%5E2%29%5E2-%284xy%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B8y%5E2%29%5E2-%284xy%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and I can re-write it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B8y%5E2%2B4xy%29%28x%5E2%2B8y%5E2-4xy%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B8y%5E2%2B4xy%29%28x%5E2%2B8y%5E2-4xy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For 2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2By%5E2%2B25 ALIGN=MIDDLE  ALT=\"x%5E4%2By%5E2%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and 3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2By-11x%5E2y%5E2%2By%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2By-11x%5E2y%5E2%2By%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , I see some squares, but I do not immediately see what to do with them.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2By%5E2%2B25 ALIGN=MIDDLE  ALT=\"x%5E4%2By%5E2%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > has three squares: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4=%28x%5E2%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E4=%28x%5E2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=%28y%29%5E2 ALIGN=MIDDLE  ALT=\"y%5E2=%28y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25=5%5E2 ALIGN=MIDDLE  ALT=\"25=5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2By-11x%5E2y%5E2%2By%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2By-11x%5E2y%5E2%2By%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > has the squares <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4=%28x%5E2%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E4=%28x%5E2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E4=%28y%5E2%29%5E2 ALIGN=MIDDLE  ALT=\"y%5E4=%28y%5E2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );