document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187272>Question 1187272</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One number is 10 more than a second number. If the product of the two numbers is 144, what are the two numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #818227 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=818227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One number is 10 more than a second number. If the product of the two numbers is 144, what are the two numbers?<BR>\n" );
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document.write( "Try pairs of integer factors of 144: 2*72, 3*48, etc.<BR>\n" );
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document.write( "To solve it algebraically,<BR>\n" );
document.write( "x*(x+10) = 144<BR>\n" );
document.write( "x^2 + 10x - 144 = 0<BR>\n" );
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document.write( "Now, to factor it, you need to find 2 numbers that differ by 10 and have a product of 144.<BR>\n" );
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document.write( "If they were not integers, you would have to use the quadratic equation or complete the square.<BR>\n" );
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document.write( "eg:<BR>\n" );
document.write( "One number is 10 more than a second number. If the product of the two numbers is 154, what are the two numbers?<BR>\n" );
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document.write( "x^2 + 10x - 154 = 0<BR>\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=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</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%2B10x%2B-154+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B10x%2B-154+=+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=%2810%29%5E2-4%2A1%2A-154=716 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2810%29%5E2-4%2A1%2A-154=716\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=716 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+=+%28-10%2B-sqrt%28+716+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-10%2B-sqrt%28+716+%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-%2810%29%2Bsqrt%28+716+%29%29%2F2%5C1+=+8.37908816025965 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2810%29%2Bsqrt%28+716+%29%29%2F2%5C1+=+8.37908816025965\"   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-%2810%29-sqrt%28+716+%29%29%2F2%5C1+=+-18.3790881602597 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2810%29-sqrt%28+716+%29%29%2F2%5C1+=+-18.3790881602597\"   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%2B10x%2B-154 ALIGN=MIDDLE  ALT=\"1x%5E2%2B10x%2B-154\"   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%2B10x%2B-154+=+%28x-8.37908816025965%29%2A%28x--18.3790881602597%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B10x%2B-154+=+%28x-8.37908816025965%29%2A%28x--18.3790881602597%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 8.37908816025965, -18.3790881602597.\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%2B10%2Ax%2B-154+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B10%2Ax%2B-154+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The solver always says it can be factored, but not necessarily with integers.<BR>\n" );
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