document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=51433>Question 51433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find two integers whose sum is 96 and whose product is 1728. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #34376 by </B><A HREF=/tutors/aboutme.mpl?userid=rchill><B>rchill(405)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rchill><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=34376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x and y represent the two integers.  We now have equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=96 ALIGN=MIDDLE  ALT=\"x%2By=96\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy=1728 ALIGN=MIDDLE  ALT=\"xy=1728\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Using the second equation, let's solve for y by dividing both sides by x:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy%29%2Fx=1728%2Fx ALIGN=MIDDLE  ALT=\"%28xy%29%2Fx=1728%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1728%2Fx ALIGN=MIDDLE  ALT=\"y=1728%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now use this value of y and substitute it into the first equation thusly:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1728%2Fx=96 ALIGN=MIDDLE  ALT=\"x%2B1728%2Fx=96\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Multiplying both sides by x produces <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B1728=96x ALIGN=MIDDLE  ALT=\"x%5E2%2B1728=96x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now subtract 96x from both sides to yield <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-96x%2B1728=0 ALIGN=MIDDLE  ALT=\"x%5E2-96x%2B1728=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we just have to factor this equation into (x - something) and (x - something_else), where the 'something' and 'something_else' when multiplied together produces 1728, and when added together they produce -96.<br><BR>\n" );
document.write( "The best way to do this is not by trial and error, but by using the discriminant.  <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=1x%5E2%2B-96x%2B1728+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-96x%2B1728+=+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=%28-96%29%5E2-4%2A1%2A1728=2304 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-96%29%5E2-4%2A1%2A1728=2304\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=2304 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--96%2B-sqrt%28+2304+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--96%2B-sqrt%28+2304+%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-%28-96%29%2Bsqrt%28+2304+%29%29%2F2%5C1+=+72 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-96%29%2Bsqrt%28+2304+%29%29%2F2%5C1+=+72\"   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-%28-96%29-sqrt%28+2304+%29%29%2F2%5C1+=+24 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-96%29-sqrt%28+2304+%29%29%2F2%5C1+=+24\"   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=1x%5E2%2B-96x%2B1728 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-96x%2B1728\"   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=1x%5E2%2B-96x%2B1728+=+1%28x-72%29%2A%28x-24%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-96x%2B1728+=+1%28x-72%29%2A%28x-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 72, 24.\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+1%2Ax%5E2%2B-96%2Ax%2B1728+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-96%2Ax%2B1728+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );