document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206593>Question 206593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156149 by </B><A HREF=/tutors/aboutme.mpl?userid=Marth><B>Marth(57)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Marth><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=156149\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the width of the grid.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B24 ALIGN=MIDDLE  ALT=\"x%5E2+%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the number of characters<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2+-25 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2+-25\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the number of characters.<BR>\n" );
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document.write( "You can solve this algebraically, or using properties of squares, which is faster.<BR>\n" );
document.write( "For any two numbers <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B1+=+b ALIGN=MIDDLE  ALT=\"a%2B1+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-a%5E2=a%2Bb ALIGN=MIDDLE  ALT=\"b%5E2-a%5E2=a%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Here we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B24=%28x%2B1%29%5E2-25 ALIGN=MIDDLE  ALT=\"x%5E2%2B24=%28x%2B1%29%5E2-25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2B25+=+%28x%2B1%29%5E2-x%5E2 ALIGN=MIDDLE  ALT=\"24%2B25+=+%28x%2B1%29%5E2-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which matches the difference of two squares above.<BR>\n" );
document.write( "So, x=24 and x+1 = 25.<BR>\n" );
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document.write( "To find the number of characters, use either value in the equations obtained at the start.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B24 ALIGN=MIDDLE  ALT=\"x%5E2+%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the number of characters<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%5E2+%2B+24 ALIGN=MIDDLE  ALT=\"24%5E2+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576+%2B+24 ALIGN=MIDDLE  ALT=\"576+%2B+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=600 </SPAN>\n" );
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