document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120184>Question 120184</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is a factoring word problem when you have to use the property of zero. The problem says \"A garden that is 4 meters wide and 6 meters long is to have a uniform border such that the area of the border is the same as the area of the garden. Find the width of the border.Could you please help me? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88072 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=88072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's say that the width of the border is x.  Then the overall width of the garden and its border on both sides must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2x ALIGN=MIDDLE  ALT=\"4%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Likewise, the overall length must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B2x ALIGN=MIDDLE  ALT=\"6%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  See the diagram.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-10%2C10%2C-10%2C10%2C%0D%0Aline%28-4%2C6%2C4%2C6%29%2C%0D%0Aline%28-4%2C6%2C-4%2C-6%29%2C%0D%0Aline%284%2C6%2C4%2C-6%29%2C%0D%0Aline%28-4%2C-6%2C4%2C-6%29%2C%0D%0Agreen%28line%28-6%2C8%2C6%2C8%29%2C%0D%0Aline%28-6%2C8%2C-6%2C-8%29%2C%0D%0Aline%286%2C8%2C6%2C-8%29%2C%0D%0Aline%28-6%2C-8%2C6%2C-8%29%29%2C%0D%0Ared%28locate%28-1.5%2C-5%2C4%2Ameters%29%2C%0D%0Alocate%28-3.5%2C0%2C6%2Ameters%29%2C%0D%0Alocate%28-.4%2C-6.6%2Cx%29%2C%0D%0Alocate%28-5.4%2C0%2Cx%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the dimensions of the outer edge of the border are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2x ALIGN=MIDDLE  ALT=\"4%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B2x ALIGN=MIDDLE  ALT=\"6%2B2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since the area of the garden is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A6=24m%5E2 ALIGN=MIDDLE  ALT=\"4%2A6=24m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the area of the border has to be equal to that, the overall area of the garden and the border must be 2 times the area of the garden or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48m%5E2 ALIGN=MIDDLE  ALT=\"48m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  But the overall area is also given by its length times its width, so we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2B2x%29%286%2B2x%29=48 ALIGN=MIDDLE  ALT=\"%284%2B2x%29%286%2B2x%29=48\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use FOIL:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2B20x%2B4x%5E2=48 ALIGN=MIDDLE  ALT=\"24%2B20x%2B4x%5E2=48\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Put into standard form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B20x-24=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B20x-24=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Divide by 4:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B5x-6=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B5x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Factors of -6 that add to +5 are 6 and -1, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x%2B6%29=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%2B6%29=0\"   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=%28x-1%29%28x%2B6%29=0 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%2B6%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but you can exclude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > because we are looking for a length measurement.\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1m ALIGN=MIDDLE  ALT=\"x=1m\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "The outside dimensions of the border must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2%2A1=6 ALIGN=MIDDLE  ALT=\"4%2B2%2A1=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B2%2A1=8 ALIGN=MIDDLE  ALT=\"6%2B2%2A1=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the overall area must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A8=48m%5E2 ALIGN=MIDDLE  ALT=\"6%2A8=48m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract the area of the original garden, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24m%5E2 ALIGN=MIDDLE  ALT=\"24m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the area of the border is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48-24=24m%5E2 ALIGN=MIDDLE  ALT=\"48-24=24m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  All conditions of the problem are met:  The border is uniform, that is it is 1 meter wide on all sides, and the area of the border is equal to the area of the garden.  Answer checks.\r<BR>\n" );
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document.write( "Hope that helps,<BR>\n" );
document.write( "John </SPAN>\n" );
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