document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=903411>Question 903411</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two workmen can paint a house together in 6 and 1/3 days. in how many days can each paint it alone if it takes one of them two days longer than the other? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #547990 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=547990\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1/x+1/(x+2)=1/6.333<BR>\n" );
document.write( "(6.333*x*(x+2))/x+(6.333*x*(x+2))/(x+2)=(6.333*x*(x+2))/6.333<BR>\n" );
document.write( "(6.333*(x+2))+6.333*x=x*(x+2)<BR>\n" );
document.write( "6.333*x+12.666+6.333*x=x^2+2*x<BR>\n" );
document.write( "12.666*x+12.666=x^2+2*x<BR>\n" );
document.write( "x^2+-10.666*x-12.666=0<BR>\n" );
document.write( "<TABLE  BORDER=1>\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-10.666x%2B-12.666=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-10.666x%2B-12.666=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-10.666x%2B-12.666=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-10.666x%2B-12.666=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+-10.666%29+%2A+x%2B-12.666=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight_red%28+-10.666%29+%2A+x%2B-12.666=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of x is -10.666, we know that -10.666=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+-10.666%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+-10.666%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-10.666%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-10.666%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-10.666%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-10.666%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-10.666x%2Bhighlight_green%28+-12.666+%29=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-10.666x%2Bhighlight_green%28+-12.666+%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-10.666%2F2%29%5E2+%2B+othernumber+=+x%5E2%2B-10.666x%2Bhighlight_red%28+-10.666%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 -12.666 (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=-10.666%5E2%2F4+%2B+othernumber+=+-12.666 ALIGN=MIDDLE  ALT=\"-10.666%5E2%2F4+%2B+othernumber+=+-12.666\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+-12.666--10.666%5E2%2F4+=+-41.106889 ALIGN=MIDDLE  ALT=\"othernumber+=+-12.666--10.666%5E2%2F4+=+-41.106889\"   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-10.666%2F2%29%5E2+%2B+-41.106889+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B-10.666%2F2%29%5E2+%2B+-41.106889+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-10.666%2F2%29%5E2+=+41.106889 ALIGN=MIDDLE  ALT=\"%28x%2B-10.666%2F2%29%5E2+=+41.106889\"   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-10.666%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B-10.666%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (41.106889).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Since the right part 41.106889 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-10.666%2F2%29+=+%2Bsqrt%28+41.106889+%29%2C+%28x%2B-10.666%2F2%29+=+-sqrt%28+41.106889+%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-10.666%2F2%29+=+6.41146543311278%2C+%28x%2B-10.666%2F2%29+=+-6.41146543311278+%29 ALIGN=MIDDLE   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-10.666%2F2+=+6.41146543311278%2C+x%2B-10.666%2F2+=+-6.41146543311278+%29 ALIGN=MIDDLE  ALT=\"system%28+x%2B-10.666%2F2+=+6.41146543311278%2C+x%2B-10.666%2F2+=+-6.41146543311278+%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+=+6.41146543311278--10.666%2F2%2C+x+=+-6.41146543311278--10.666%2F2+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+6.41146543311278--10.666%2F2%2C+x+=+-6.41146543311278--10.666%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+=+11.7444654331128%2C+x+=+-1.07846543311278+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+11.7444654331128%2C+x+=+-1.07846543311278+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: x=11.7444654331128, -1.07846543311278.\n" );
document.write( "</TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "11.7444654331128 days<BR>\n" );
document.write( "13.7444654331128 days<BR>\n" );
document.write( "check<BR>\n" );
document.write( "a=11.7444654 days<BR>\n" );
document.write( "b=13.7444654 days<BR>\n" );
document.write( "1/a+1/b=1/x<BR>\n" );
document.write( "1/11.7444654+1/13.7444654=1/x<BR>\n" );
document.write( "1/11.7444654+1/13.7444654=1/6.333<BR>\n" );
document.write( "1/x=1/6.333<BR>\n" );
document.write( "x=6.333 days working together\r<BR>\n" );
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document.write( " </SPAN>\n" );
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