document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=663432>Question 663432</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My teacher put this problem and solution on the board and I need a laymans termed explaination of why he did certain things in the solution.\r<BR>\n" );
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document.write( "3x^2+4y^2-6x-24y+39=0<BR>\n" );
document.write( "3(x^2-2x+1)+4(y^2-6y+1)=0-39+3+36<BR>\n" );
document.write( "3(x-1)^2+4(y-3)^2=0\r<BR>\n" );
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document.write( "So, I get confused when my teacher added 1 inside both of the parenthesis...why did he do that?  Where did he get the 1 from? Why not just leave the 1 out of both of the parenthesis?  Please help!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #412778 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=412778\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "He didn't add 1 inside of both sets of parentheses, unless he just made a mistake that is.  However, the 36 term added to the RHS tells me otherwise.\r<BR>\n" );
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document.write( "What he is doing is completing the square on each of the variables.  The goal, when you have a general form equation of an ellipse like this is to convert it to the standard form, namely:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{(x\ -\ h)^2}{a^2}\ +\ \frac{(y\ -\ k)^2}{b^2}\ =\ 1\"> which is a ellipse centered at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,\,k)\"> with semi-major and semi-minor axes <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\"> (the major axis is larger).\r<BR>\n" );
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document.write( "To perform this conversion, we have to complete the square.\r<BR>\n" );
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document.write( "Step 1:  Move the constant term into the RHS:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3x^2\ +\ 4y^2\ -\ 6x\ -\ 24y\ =\ -39\">\r<BR>\n" );
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document.write( "Step 2:  Put the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> terms together and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> terms together\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3x^2\ -\ 6x\ +\ 4y^2\ -\ 24y\ =\ -39\">\r<BR>\n" );
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document.write( "Step 3:  Factor the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2\"> term out of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> terms, leaving a spot inside the parentheses, then do the same thing for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3\left(x^2\ -\ 2x\ +\ ??\right)\ +\ 4\left(y^2\ -\ 6y\ +\ ??\right)\ =\ -\ 39\">\r<BR>\n" );
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document.write( "Step 4: Complete the square on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">.  Divide the coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> by 2, and square the result.  -2 divided by 2 is -1, -1 squared is 1, and therefore 1 is the number that needs to be added inside of the parentheses.  Note that the 3 coefficient on the outside of the parentheses means that you are actually adding 3 to the LHS when you insert a 1 into the set of parentheses containing the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> term.  Hence you have to compensate by adding 3 to the RHS.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3\left(x^2\ -\ 2x\ +\ 1\right)\ +\ 4\left(y^2\ -\ 6y\ +\ ??\right)\ =\ -\ 39\ +\ 3\">\r<BR>\n" );
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document.write( "Step 5: Do the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> part.  -6 divided by 2 is -3.  -3 squared is 9. 9 inside the parentheses and 9 times 4 = 36 in the RHS:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3\left(x^2\ -\ 2x\ +\ 1\right)\ +\ 4\left(y^2\ -\ 6y\ +\ 9\right)\ =\ -\ 39\ +\ 3\ +\ 36\">\r<BR>\n" );
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document.write( "Step 6:\r<BR>\n" );
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document.write( "Factor the expressions in the parentheticals, which should be easy to do because we arranged for them to be perfect squares.  Also collect terms in the RHS.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3\left(x\ -\ 1\right)^2\ +\ 4\left(y\ -\ 3\right)^2\ =\ 0\">\r<BR>\n" );
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document.write( "Step 7:\r<BR>\n" );
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document.write( "Gaack!  The RHS is zero!  We won't have an ellipse, we will have a point.  Be that as it may, your RHS might not be zero next time.  Proceed:\r<BR>\n" );
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document.write( "Ordinarily you would divide by the simplified RHS to produce a 1 in the RHS, but that is not to be in this case. Divide by the LCM of the coefficients on the parenthetical expressions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\left(x\ -\ 1\right)^2}{4}\ +\ \frac{\left(y\ -\ 3\right)^2}{3}\ =\ 0\">\r<BR>\n" );
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document.write( "And enjoy your graph of a point.  By the way:  Super Double WoWWie Extra Credit.  What are the coordinates of that point?\r<BR>\n" );
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document.write( "Step 8:  Don't doze off in class any more.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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