document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=705702>Question 705702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two numbers differ by 8, their product is 65,what is the number?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #434728 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39618)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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=434728\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Some may quickly recognize that as a product, and being a factor of 5, 13*5=65, which we may also recognize as 12*5 plus one more 5.  Basic addition fact may be in some people's heads and 8+5=13 is immediately recognized.  \r<BR>\n" );
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document.write( "You ask for \"what's the number\", but the description indicates TWO numbers.  I see 13 and 5.  \r<BR>\n" );
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document.write( "You probably want an algebraic arithmetic treatment so here it is.\r<BR>\n" );
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document.write( "Numbers m and n.<BR>\n" );
document.write( "Differ by 8:  n-m=8<BR>\n" );
document.write( "Product is 65:  mn=65<BR>\n" );
document.write( "Find m and n.\r<BR>\n" );
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document.write( "The Differ equation gives n=m+8.  Use this in the Product equation.<BR>\n" );
document.write( "m(m+8)=65<BR>\n" );
document.write( "m^2+8m-65=0, which we can factor, so let's do that:<BR>\n" );
document.write( "(m+5)(m-13) ---- will that work?  No.  Gives -8m instead of +8m.<BR>\n" );
document.write( "(m-5)(m+13) ---- Will that work?  Yes.  13m-5m=8m, and product in const. term is -65.\r<BR>\n" );
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document.write( "OKAY, solution to the quadratic formed is m=5 or m=-13.  <BR>\n" );
document.write( "What about n?<BR>\n" );
document.write( "n-m=8<BR>\n" );
document.write( "n=m+8<BR>\n" );
document.write( "For m=5, n=13.<BR>\n" );
document.write( "For m=-13, n=-5.\r<BR>\n" );
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document.write( "ANSWER: Both solutions fit the description.  Either the numbers are -5 and -13, or they are 5 and 13.   </SPAN>\n" );
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