document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=437040>Question 437040</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The difference of two numbers is 1. What is the smallest possible value for the sum of their squares? Find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #302308 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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=302308\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x = one number<BR>\n" );
document.write( "y = a number that is one less than x<BR>\n" );
document.write( "y = x-1<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The sum of squares =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+%28x-1%29%5E2+=+x%5E2+%2B+x%5E2+-2x+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+%28x-1%29%5E2+=+x%5E2+%2B+x%5E2+-2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x%5E2+-2x+%2B+1+=+2x%5E2+-2x+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x%5E2+-2x+%2B+1+=+2x%5E2+-2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Assuming you mean 'integers' when you say 'numbers', then we could try some values:<BR>\n" );
document.write( "When x = 0,  the sum of squares = 1<BR>\n" );
document.write( "When x = -1, the sum of squares = 4<BR>\n" );
document.write( "When x = 1, the sum of squares = 1<BR>\n" );
document.write( "...<BR>\n" );
document.write( "But randomly plugging in values of x and checking the sum is not a very mathematical approach.<BR>\n" );
document.write( "Instead, let's look at the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%2Ax%2Ax+-+2%2Ax+%2B+1%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%2Ax%2Ax+-+2%2Ax+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We immediately see this is a parabola and that the minimum value occurs when x = 1/2.<BR>\n" );
document.write( "Checking this value we find:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-2x+%2B+1+=+2%281%2F4%29+-2%281%2F2%29+%2B+1 ALIGN=MIDDLE  ALT=\"2x%5E2+-2x+%2B+1+=+2%281%2F4%29+-2%281%2F2%29+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%281%2F4%29+-2%281%2F2%29+%2B+1+=+1%2F2+-1+%2B+1 ALIGN=MIDDLE  ALT=\"2%281%2F4%29+-2%281%2F2%29+%2B+1+=+1%2F2+-1+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2+-1+%2B+1+=+1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2+-1+%2B+1+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking back at the question it asks for two numbers<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2 ALIGN=MIDDLE  ALT=\"x+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1+=+-1%2F2 ALIGN=MIDDLE  ALT=\"x-1+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "What is their sum of their squares?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F4+%2B+1%2F4+=+1%2F2 ALIGN=MIDDLE  ALT=\"1%2F4+%2B+1%2F4+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That fits with our graph, too.<BR>\n" );
document.write( "Answer: The two numbers are 1/2 and -1/2.<BR>\n" );
document.write( "Done. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );