document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=617244>Question 617244</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ab +bc+ ca , when a + b + c+12 and a*a = b*b +c*c =50 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #388266 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=388266\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I believe you had a typo in your question. Maybe you had to evaluate ab + bc + ca, and were given as information that<BR>\n" );
document.write( "a + b + c = 12 and a*a = b*b +c*c =50<BR>\n" );
document.write( "(I'll write that last equation as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aa=b%2Ab%2Bc%2Ac=50 ALIGN=MIDDLE  ALT=\"a%2Aa=b%2Ab%2Bc%2Ac=50\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( " <BR>\n" );
document.write( "If that was the problem, we could start with the product <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%2Bc%29%28a%2Bb%2Bc%29 ALIGN=MIDDLE  ALT=\"%28a%2Bb%2Bc%29%28a%2Bb%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and apply the distributive property twice to get <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%2Bc%29%28a%2Bb%2Bc%29=a%28a%2Bb%2Bc%29%2Bb%28a%2Bb%2Bc%29=c%28a%2Bb%2Bc%29=a%2Aa%2Ba%2Ab%2Ba%2Ac%2Bb%2Aa%2Bb%2Ab%2Bb%2Ac%2Bc%2Aa%2Bc%2Ab%2Bc%2Ac ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To simplify the writing, I would like to start omitting some of those asterisks and middle dot symbols for multiplication,<BR>\n" );
document.write( "to be able to write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab ALIGN=MIDDLE  ALT=\"ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of a*b or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ab ALIGN=MIDDLE  ALT=\"a%2Ab\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bc ALIGN=MIDDLE  ALT=\"bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Ac ALIGN=MIDDLE  ALT=\"b%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the like.<BR>\n" );
document.write( "I would also apply the commutative property that says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab=a%2Ab=+b%2Aa=ba ALIGN=MIDDLE  ALT=\"ab=a%2Ab=+b%2Aa=ba\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and similarly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ac=ca ALIGN=MIDDLE  ALT=\"ac=ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=bc=cb ALIGN=MIDDLE  ALT=\"bc=cb\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so that I can write all those products as the ab, bc, or ca in the formula that needs to be evaluated<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aa%2Ba%2Ab%2Ba%2Ac%2Bb%2Aa%2Bb%2Ab%2Bb%2Ac%2Bc%2Aa%2Bc%2Ab%2Bc%2Ac=a%2Aa%2Bab%2Bca%2Bab%2Bb%2Ab%2Bbc%2Bca%2Bbc%2Bc%2Ac ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next, I would rearrange the terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aa%2Bab%2Bca%2Bab%2Bb%2Ab%2Bbc%2Bca%2Bbc%2Bc%2Ac=a%2Aa%2Bb%2Ab%2Bc%2Ac%2Bab%2Bab%2Bbc%2Bbc%2Bca%2Bca ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I would also group them to get together similar terms.<BR>\n" );
document.write( "(I also want to group together the expressions that the problem gives values for).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aa%2Bb%2Ab%2Bc%2Ac%2Bab%2Bab%2Bbc%2Bbc%2Bca%2Bca=a%2Aa%2B%28b%2Ab%2Bc%2Ac%29%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we have that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%2Bc%29%28a%2Bb%2Bc%29=a%2Aa%2B%28b%2Ab%2Bc%2Ac%29%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next, we replace the known values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Aa ALIGN=MIDDLE  ALT=\"a%2Aa\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Ab%2Bc%2Ac ALIGN=MIDDLE  ALT=\"b%2Ab%2Bc%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A12=50%2B50%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29 ALIGN=MIDDLE  ALT=\"12%2A12=50%2B50%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144=100%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29 ALIGN=MIDDLE  ALT=\"144=100%2B%28%28ab%2Bbc%2Bca%29%2B%28ab%2Bbc%2Bca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144=100%2B2%28ab%2Bbc%2Bca%29 ALIGN=MIDDLE  ALT=\"144=100%2B2%28ab%2Bbc%2Bca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point, I first subtract 100 from the expressions on both sides of the equal sign:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144=100%2B2%28ab%2Bbc%2Bca%29 ALIGN=MIDDLE  ALT=\"144=100%2B2%28ab%2Bbc%2Bca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144-100=100%2B2%28ab%2Bbc%2Bca%29-100 ALIGN=MIDDLE  ALT=\"144-100=100%2B2%28ab%2Bbc%2Bca%29-100\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=44=2%28ab%2Bbc%2Bca%29 ALIGN=MIDDLE  ALT=\"44=2%28ab%2Bbc%2Bca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, I divide the expressions on both sides of the equal sign by 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=44=2%28ab%2Bbc%2Bca%29 ALIGN=MIDDLE  ALT=\"44=2%28ab%2Bbc%2Bca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=44%2F2=2%28ab%2Bbc%2Bca%29%2F2 ALIGN=MIDDLE  ALT=\"44%2F2=2%28ab%2Bbc%2Bca%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22=ab%2Bbc%2Bca ALIGN=MIDDLE  ALT=\"22=ab%2Bbc%2Bca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28ab%2Bbc%2Bca=22%29 ALIGN=MIDDLE  ALT=\"highlight%28ab%2Bbc%2Bca=22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );