document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=173445>Question 173445</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two positive numbers differ by 11, and their square roots differ by 1. 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 #128301 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=128301\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's go at this straight up and see what happens:\r<BR>\n" );
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document.write( "Let one of the numbers be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the other one be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we know that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=11 ALIGN=MIDDLE  ALT=\"x-y=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29-sqrt%28y%29=1 ALIGN=MIDDLE  ALT=\"sqrt%28x%29-sqrt%28y%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Yuck!  You can see right off that those radicals are going to make this messy.\r<BR>\n" );
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document.write( "So let's try another tack.\r<BR>\n" );
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document.write( "Let's keep <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the desired variables, but let's define two more variables, thus:\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"t=sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=sqrt%28y%29 ALIGN=MIDDLE  ALT=\"u=sqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  That means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t%5E2 ALIGN=MIDDLE  ALT=\"x=t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=u%5E2 ALIGN=MIDDLE  ALT=\"y=u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now we can write:\r<BR>\n" );
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document.write( "1.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2-u%5E2=11 ALIGN=MIDDLE  ALT=\"t%5E2-u%5E2=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > and\r<BR>\n" );
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document.write( "2.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-u=1 ALIGN=MIDDLE  ALT=\"t-u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve Eq. 2 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=u%2B1 ALIGN=MIDDLE  ALT=\"t=u%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and substitute this expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > in Eq. 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28u%2B1%29%5E2-u%5E2=11 ALIGN=MIDDLE  ALT=\"%28u%2B1%29%5E2-u%5E2=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Expand the binomial and collect terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2%2B2u%2B1-u%5E2=11 ALIGN=MIDDLE  ALT=\"u%5E2%2B2u%2B1-u%5E2=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u%2B1=11 ALIGN=MIDDLE  ALT=\"2u%2B1=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2u=10 ALIGN=MIDDLE  ALT=\"2u=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=5 ALIGN=MIDDLE  ALT=\"u=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t-u=1 ALIGN=MIDDLE  ALT=\"t-u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=6 ALIGN=MIDDLE  ALT=\"t=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t%5E2=36 ALIGN=MIDDLE  ALT=\"x=t%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=u%5E2=25 ALIGN=MIDDLE  ALT=\"y=u%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, 36 and 25 are the two numbers.\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-25=11 ALIGN=MIDDLE  ALT=\"36-25=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2836%29-sqrt%2825%29=6-5=1 ALIGN=MIDDLE  ALT=\"sqrt%2836%29-sqrt%2825%29=6-5=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer checks.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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