document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=918203>Question 918203</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The difference between two numbers is 9 and the product of the numbers is 162. Find the two numbers.<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 #558762 by </B><A HREF=/tutors/aboutme.mpl?userid=srinivas.g><B>srinivas.g(540)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=srinivas.g><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=srinivas.g><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=558762\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x , y be the 2 numbers<BR>\n" );
document.write( "  as per the data  x-y  =9<BR>\n" );
document.write( "  and   x*y = 162 <BR>\n" );
document.write( "     but x-y =9 <BR>\n" );
document.write( "       x = 9+y<BR>\n" );
document.write( " put x=  9+y in x*y = 162 <BR>\n" );
document.write( "                (9+y) *y =162<BR>\n" );
document.write( "                9*y+y*y= 162 <BR>\n" );
document.write( "               9y+y^2 =162<BR>\n" );
document.write( "                  y^2+9y =162 <BR>\n" );
document.write( " subtract 162 on both sides <BR>\n" );
document.write( "                     y^2 +9y -162 =162-162<BR>\n" );
document.write( "                       y^2+9y-162 =0<BR>\n" );
document.write( " solve the quadratic equation   <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=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=ay%5E2%2Bby%2Bc=0 ALIGN=MIDDLE  ALT=\"ay%5E2%2Bby%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y%5E2%2B9y%2B-162+=+0 ALIGN=MIDDLE  ALT=\"1y%5E2%2B9y%2B-162+=+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=y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"y%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=%289%29%5E2-4%2A1%2A-162=729 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%289%29%5E2-4%2A1%2A-162=729\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=729 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+=+%28-9%2B-sqrt%28+729+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-9%2B-sqrt%28+729+%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=y%5B1%5D+=+%28-%289%29%2Bsqrt%28+729+%29%29%2F2%5C1+=+9 ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+%28-%289%29%2Bsqrt%28+729+%29%29%2F2%5C1+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D+=+%28-%289%29-sqrt%28+729+%29%29%2F2%5C1+=+-18 ALIGN=MIDDLE  ALT=\"y%5B2%5D+=+%28-%289%29-sqrt%28+729+%29%29%2F2%5C1+=+-18\"   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=1y%5E2%2B9y%2B-162 ALIGN=MIDDLE  ALT=\"1y%5E2%2B9y%2B-162\"   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=1y%5E2%2B9y%2B-162+=+1%28y-9%29%2A%28y--18%29 ALIGN=MIDDLE  ALT=\"1y%5E2%2B9y%2B-162+=+1%28y-9%29%2A%28y--18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 9, -18.\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%2B9%2Ax%2B-162+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B9%2Ax%2B-162+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "so  y is either 9 or -18<BR>\n" );
document.write( " if y =9 then x  = 9+y=9+9=18<BR>\n" );
document.write( "  so numbers  x  =18 & y =9<BR>\n" );
document.write( " if  y =-18   then x= 9+y=9-18 =-9<BR>\n" );
document.write( "       so numbers x =-9 & y=-18 </SPAN>\n" );
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