document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=85927>Question 85927</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what two numbers have a product of 48 and a sum of 24. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62098 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=62098\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Would it help you to know that this problem does not have a \"nice\" solution?<BR>\n" );
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document.write( "Call the two unknown numbers x and y.<BR>\n" );
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document.write( "The problem tells you that the product of the two numbers is 48. Put this in equation form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay+=+48+ ALIGN=MIDDLE  ALT=\"x%2Ay+=+48+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The problem also tells you that their sum is 24. Put this in equation form also:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%2B+y+=+24 ALIGN=MIDDLE  ALT=\"+x+%2B+y+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Let's solve these two equations by variable elimination. Solve one of the two equations<BR>\n" );
document.write( "for one of the variables in terms of the other.  For example, take the first equation<BR>\n" );
document.write( "(the product equation) and solve for x in terms of y. Do this by dividing both sides of the<BR>\n" );
document.write( "equation by y.  When you do that you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ay%2Fy+=+48%2Fy ALIGN=MIDDLE  ALT=\"x%2Ay%2Fy+=+48%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On the left side the y in the denominator cancels the y in the numerator and the equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+48%2Fy ALIGN=MIDDLE  ALT=\"x+=+48%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now take the right side of this equation and substitute it into the second equation<BR>\n" );
document.write( "(the sum equation) in place of x since the right side is equal to x:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y+=+24 ALIGN=MIDDLE  ALT=\"x+%2B+y+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48%2Fy+%2B+y+=+24 ALIGN=MIDDLE  ALT=\"48%2Fy+%2B+y+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can get rid of the denominator by multiplying both sides (all terms) by y. When you<BR>\n" );
document.write( "do that the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2A48%2Fy+%2B+y%2Ay+=+24%2Ay ALIGN=MIDDLE  ALT=\"y%2A48%2Fy+%2B+y%2Ay+=+24%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Cancel the y in the denominator of the first term with the y in the numerator, and also<BR>\n" );
document.write( "recognize that y*y is y-squared. The equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48+%2B+y%5E2+=+24y ALIGN=MIDDLE  ALT=\"48+%2B+y%5E2+=+24y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 24y from both sides of the equation and it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48+%2B+y%5E2+-+24y+=+0 ALIGN=MIDDLE  ALT=\"48+%2B+y%5E2+-+24y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Arrange the terms in descending powers of y and you have the standard quadratic equation form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+-+24y+%2B+48+=+0 ALIGN=MIDDLE  ALT=\"y%5E2+-+24y+%2B+48+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Use the quadratic formula to solve for y.  The quadratic formula says that for a quadratic<BR>\n" );
document.write( "equation of the form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ay%5E2+%2B+by+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ay%5E2+%2B+by+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "the solution for y is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-b+%2B-+sqrt%28+b%5E2-%284%2Aa%2Ac%29+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"y+=+%28-b+%2B-+sqrt%28+b%5E2-%284%2Aa%2Ac%29+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By comparing the quadratic form with the equation you are trying to solve you can see<BR>\n" );
document.write( "that a = 1, b = -24, and c = 48. Substitute these values into the equation for y and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-%28-24%29+%2B-+sqrt%28%28-24%29%5E2+-+4%2A1%2A48%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"y+=+%28-%28-24%29+%2B-+sqrt%28%28-24%29%5E2+-+4%2A1%2A48%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that -(-24) equals + 24, (-24)^2 = 576, and -(4*1*48) = -192, and (2*1) = 2. Substitute <BR>\n" );
document.write( "these values into the  equation for y and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2824+%2B-+sqrt%28576-192%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%2824+%2B-+sqrt%28576-192%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now note that (576 - 192) = 384 and the square root of 384 is 19.59591794. Substitute<BR>\n" );
document.write( "this value for the radical term and the equation for y becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2824+%2B-19.59591794%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%2824+%2B-19.59591794%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This generates two possible values for y. When you have the + sign, the value of y becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2824+%2B+19.59591794%29%2F2+=+43.59591794%2F2+=+21.79795897 ALIGN=MIDDLE  ALT=\"y+=+%2824+%2B+19.59591794%29%2F2+=+43.59591794%2F2+=+21.79795897\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You know that the sum of x and y must equal 24.  So if y = 21.79795897, then x must equal:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+24+-+21.79795897+=+2.20204103 ALIGN=MIDDLE  ALT=\"x+=+24+-+21.79795897+=+2.20204103\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So one possible set of answers is y = 21.79795897 and x = 2.20204103.<BR>\n" );
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document.write( "You can check this set by adding x and y and make sure they add up to 24 and then multiply<BR>\n" );
document.write( "x and y and make sure the product is 48.<BR>\n" );
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document.write( "This answer set was based on the use of a + sign preceding the radical in the equation<BR>\n" );
document.write( "for y.  Now you need to consider the - sign in front of the radical.  When you do you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2824+-+sqrt%28576-192%29%29%2F2+ ALIGN=MIDDLE  ALT=\"y+=+%2824+-+sqrt%28576-192%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Again, the term in the radical is simplifies to 384, and the square root of 384 is 19.59591794.<BR>\n" );
document.write( "Substitute this into the equation for y and this time it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%2824+-+19.59591794%29%2F2+=+4.40408206%2F2+=+2.20204103 ALIGN=MIDDLE  ALT=\"y+=+%2824+-+19.59591794%29%2F2+=+4.40408206%2F2+=+2.20204103\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Find x by subtracting this answer from 24 and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+24+-+2.20204103+=+21.79795897 ALIGN=MIDDLE  ALT=\"x+=+24+-+2.20204103+=+21.79795897\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So this time the answer set is y = 2.20204103 and x = 21.79795897.<BR>\n" );
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document.write( "But when you compare the answer two sets of answers, you find that one of the numbers<BR>\n" );
document.write( "is 21.79795897 and the other number is 2.20204103.  <BR>\n" );
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document.write( "So the answer to this problem is just that ... the two numbers are 21.79795897 and 2.20204103.<BR>\n" );
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document.write( "As I said at the start, the answers are not very \"nice\" and the only reason they don't have<BR>\n" );
document.write( "more than 8 places after the decimal point is that is the number of places my calculator<BR>\n" );
document.write( "will handle.  If it would compute more places, there would be more than 8 decimal positions.<BR>\n" );
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document.write( "I'm not sure this will help, but it may convince you that a problem that looks so ordinary<BR>\n" );
document.write( "at the beginning can sometimes result in answers that look very odd, indeed. </SPAN>\n" );
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