document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=125713>Question 125713</A>: <I><SPAN STYLE=\"word-break: break-all;\">  complete the square, and find the roots of the quadratic equation.\r<BR>\n" );
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document.write( "  x2 - 2/3x = 0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #92101 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=92101\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+%282%2F3%29x+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+%282%2F3%29x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F3 ALIGN=MIDDLE  ALT=\"-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.66 ALIGN=MIDDLE  ALT=\"-0.66\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then you have:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=complete_square.solver>COMPLETING THE SQUARE solver for quadratics</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Read <A  rel=nofollow HREF=Fool-Proof-Completing-the-Square-Part-I.lesson>this lesson on completing the square by prince_abubu</A>, if you do not know how to complete the square.<BR>Let's convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-0.66x%2B0=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-0.66x%2B0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to standard form by dividing both sides by 1: <BR>\n" );
document.write( "We have: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-0.66x%2B0=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-0.66x%2B0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \n" );
document.write( "What we want to do now is to change this equation to a complete square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >. How can we find out values of <B>somenumber</B> and <B>othernumber</B> that would make it work?<BR>\n" );
document.write( "Look at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the coefficient in our equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2Bhighlight_red%28+-0.66%29+%2A+x%2B0=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight_red%28+-0.66%29+%2A+x%2B0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of x is -0.66, we know that -0.66=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+-0.66%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+-0.66%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, we know that our equation can be rewritten as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-0.66%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we do not yet know the other number.<BR>\n" );
document.write( "We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-0.66%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to our original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-0.66x%2Bhighlight_green%28+0+%29=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-0.66x%2Bhighlight_green%28+0+%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2+%2B+othernumber+=+x%5E2%2B-0.66x%2Bhighlight_red%28+-0.66%5E2%2F4%2Bothernumber+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  The highlighted <FONT COLOR=red>red part</FONT> must be equal to 0 (highlighted <FONT COLOR=green>green part</FONT>).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.66%5E2%2F4+%2B+othernumber+=+0 ALIGN=MIDDLE  ALT=\"-0.66%5E2%2F4+%2B+othernumber+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+0--0.66%5E2%2F4+=+-0.1089 ALIGN=MIDDLE  ALT=\"othernumber+=+0--0.66%5E2%2F4+=+-0.1089\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, the equation converts to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2+%2B+-0.1089+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B-0.66%2F2%29%5E2+%2B+-0.1089+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2+=+0.1089 ALIGN=MIDDLE  ALT=\"%28x%2B-0.66%2F2%29%5E2+=+0.1089\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Our equation converted to a square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-0.66%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B-0.66%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (0.1089).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Since the right part 0.1089 is greater than zero, there are two solutions:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-0.66%2F2%29+=+%2Bsqrt%28+0.1089+%29%2C+%28x%2B-0.66%2F2%29+=+-sqrt%28+0.1089+%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    , or<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-0.66%2F2%29+=+0.33%2C+%28x%2B-0.66%2F2%29+=+-0.33+%29 ALIGN=MIDDLE  ALT=\"system%28+%28x%2B-0.66%2F2%29+=+0.33%2C+%28x%2B-0.66%2F2%29+=+-0.33+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%2B-0.66%2F2+=+0.33%2C+x%2B-0.66%2F2+=+-0.33+%29 ALIGN=MIDDLE  ALT=\"system%28+x%2B-0.66%2F2+=+0.33%2C+x%2B-0.66%2F2+=+-0.33+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x+=+0.33--0.66%2F2%2C+x+=+-0.33--0.66%2F2+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+0.33--0.66%2F2%2C+x+=+-0.33--0.66%2F2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x+=+0.66%2C+x+=+0+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+0.66%2C+x+=+0+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: x=0.66, 0.\n" );
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