document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=689851>Question 689851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The digit of a two-digit number adds up to 14. the number itself is 11 less than the sum of the squares of the digits. find the number\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: I worked out that the two numbers are 5 and 9 by random guessing and substituting the numbers into the question. but I don't know the correct way to set out the working.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #426070 by </B><A HREF=/tutors/aboutme.mpl?userid=vleith><B>vleith(2983)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=vleith><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=426070\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the tens digit be t and the ones digit be 1 .<BR>\n" );
document.write( "You are told <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%2B+o+=+14 ALIGN=MIDDLE  ALT=\"t+%2B+o+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+14+-+o ALIGN=MIDDLE  ALT=\"t+=+14+-+o\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10t+%2B+o+=+t%5E2+%2B+o%5E2++-+11 ALIGN=MIDDLE  ALT=\"10t+%2B+o+=+t%5E2+%2B+o%5E2++-+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2A%2814+-+o%29+%2B+o+=+%2814-o%29%5E2+%2B+o%5E2+-+11 ALIGN=MIDDLE  ALT=\"10%2A%2814+-+o%29+%2B+o+=+%2814-o%29%5E2+%2B+o%5E2+-+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=140+-+9o+=+196+-+28o+%2B+o%5E2+%2B+o%5E2+-+11 ALIGN=MIDDLE  ALT=\"140+-+9o+=+196+-+28o+%2B+o%5E2+%2B+o%5E2+-+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+2o%5E2+-+19o+%2B+45 ALIGN=MIDDLE  ALT=\"0+=+2o%5E2+-+19o+%2B+45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B-19x%2B45+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-19x%2B45+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-19%29%5E2-4%2A2%2A45=1 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-19%29%5E2-4%2A2%2A45=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--19%2B-sqrt%28+1+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--19%2B-sqrt%28+1+%29%29%2F2%5Ca\"   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=x%5B1%5D+=+%28-%28-19%29%2Bsqrt%28+1+%29%29%2F2%5C2+=+5 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-19%29%2Bsqrt%28+1+%29%29%2F2%5C2+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-19%29-sqrt%28+1+%29%29%2F2%5C2+=+4.5 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-19%29-sqrt%28+1+%29%29%2F2%5C2+=+4.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B-19x%2B45 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-19x%2B45\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B-19x%2B45+=+2%28x-5%29%2A%28x-4.5%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-19x%2B45+=+2%28x-5%29%2A%28x-4.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 5, 4.5.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-19%2Ax%2B45+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-19%2Ax%2B45+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "So the only integer answer is o = 5. Which means t = 14-5 = 9 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );