document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=788876>Question 788876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two numbers have a difference of 24 . Find the numbers if the result of adding their sum and their product is a minimum. \r<BR>\n" );
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document.write( "Can you please help me out? Thanks so much in advance:)<BR>\n" );
document.write( "Can you also please show the steps it would really help me understand:) Thank-you  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #478678 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=478678\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the smallest of the two numbers<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B24 ALIGN=MIDDLE  ALT=\"x%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the other number<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bx%2B24 ALIGN=MIDDLE  ALT=\"x%2Bx%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >= their sum<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B24%29 ALIGN=MIDDLE  ALT=\"x%28x%2B24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= their product<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29 ALIGN=MIDDLE  ALT=\"y=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the result of adding their sum and their product.<BR>\n" );
document.write( "We need to find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > that minimizes the y value for that function.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Let's simplify that expression:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29 ALIGN=MIDDLE  ALT=\"y=%28x%2Bx%2B24%29%2B%28x%28x%2B24%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282x%2B24%29%2B%28x%5E2%2B24x%29 ALIGN=MIDDLE  ALT=\"y=%282x%2B24%29%2B%28x%5E2%2B24x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B24%2Bx%5E2%2B24x%7D%7D%0D%0A%7B%7B%7By=x%5E2%2B26x%2B24%7D%7D%0D%0AThat+is+a+quadratic+function+with+a+minimum.+%28The+graph+is+a+parabola%29.%0D%0ATo+find+the+coordinates+of+the+minimum+we+can+apply+formulas+taught+in+class%2C+or+we+can+just+complete+the+square.%0D%0A+%0D%0ACOMPLETING+THE+SQUARE%3A%0D%0ANo+need+to+memorize+formulas+%28unless+your+teacher+thinks+you+should%29.%0D%0A%7B%7B%7By=x%5E2%2B26x%2B24 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2B26x%2B169-169%2B24 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B26x%2B169-169%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B13%29%5E2-169%2B24 ALIGN=MIDDLE  ALT=\"y=%28x%2B13%29%5E2-169%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B13%29%5E2-145 ALIGN=MIDDLE  ALT=\"y=%28x%2B13%29%5E2-145\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B13%29%5E2%3E0 ALIGN=MIDDLE  ALT=\"%28x%2B13%29%5E2%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-13 ALIGN=MIDDLE  ALT=\"x%3C%3E-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B13%29%5E2=0 ALIGN=MIDDLE  ALT=\"%28x%2B13%29%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-13 ALIGN=MIDDLE  ALT=\"x=-13\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B13%29%5E2-145%3E-145 ALIGN=MIDDLE  ALT=\"y=%28x%2B13%29%5E2-145%3E-145\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-13 ALIGN=MIDDLE  ALT=\"x%3C%3E-13\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-13%29%5E2-145=-145 ALIGN=MIDDLE  ALT=\"y=%28x-13%29%5E2-145=-145\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the minimum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-13%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%28-13%29 ALIGN=MIDDLE  ALT=\"x=highlight%28-13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B24=-13%2B24=highlight%2811%29 ALIGN=MIDDLE  ALT=\"x%2B24=-13%2B24=highlight%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the two numbers we were looking for.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH FORMULAS:<BR>\n" );
document.write( "THe formulas you were given may be slightly different.<BR>\n" );
document.write( "You were probably given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > as general formula for the quadratic function, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the formula to find the x-coordinate of the minimum.<BR>\n" );
document.write( "In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=26 ALIGN=MIDDLE  ALT=\"b=26\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-26%2F%282%2A1%29=-26%2F2=highlight%28-13%29 ALIGN=MIDDLE  ALT=\"x=-26%2F%282%2A1%29=-26%2F2=highlight%28-13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the x-coordinate of the minimum.<BR>\n" );
document.write( "The two numbers are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%28-13%29 ALIGN=MIDDLE  ALT=\"x=highlight%28-13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B24=-13%2B24=highlight%2811%29 ALIGN=MIDDLE  ALT=\"x%2B24=-13%2B24=highlight%2811%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );