document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179483>Question 1179483</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An orchard has 728 pear trees.  The number of rows exceeds the number of trees per row by 2.  How many trees are there in each row?\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 #809059 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=809059\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Note in the solution from the other tutor how the standard algebraic solution involves factoring the quadratic equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2x-728=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B2x-728=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Few people are going to look at that and see that the factorization is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B28%29%28x-26%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B28%29%28x-26%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In their solution, they solved the equation using the quadratic formula.  That always works; but in this case it still involved working with large numbers.<br><BR>\n" );
document.write( "This is an example of a problem where a bit of insight (or perhaps a lot of experience) can make solving the problem much easier.<br><BR>\n" );
document.write( "Here is what I saw almost immediately on reading the problem: we have two numbers that differ by 2 whose product is 728.  That is two numbers that are very close together, so the 728 is very close to a perfect square.  So instead of calling the numbers x and x+2....<br><BR>\n" );
document.write( "let x-1 be the smaller number (number of trees per row)<BR>\n" );
document.write( "let x+1 be the larger number (number of rows of trees)<br><BR>\n" );
document.write( "Then the given information yields an equation that is solved relatively easily:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x%2B1%29=728 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%2B1%29=728\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-1=728 ALIGN=MIDDLE  ALT=\"x%5E2-1=728\"   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=729 ALIGN=MIDDLE  ALT=\"x%5E2=729\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=27 ALIGN=MIDDLE  ALT=\"x=27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: The number of trees in each row is x-1=26<br><BR>\n" );
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