document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=165121>Question 165121</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.	A rectangular garden has dimensions of 18 feet by 13 feet. A gravel path of uniform width is to be built around the garden. How wide can the path be if there is enough gravel for 516 square feet? \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 #121728 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=121728\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Draw a diagram of the problem -- it'll help you see how to solve it.<BR>\n" );
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document.write( "If you do, the \"inner rectangle\" represents the garden while the \"outer rectangle\" (bordered by the outside edge of the gravel path).<BR>\n" );
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document.write( "\"outer rectangle\" - \"inner rectangle\" = \"area of gravel path\"<BR>\n" );
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document.write( "Let x = width of gravel path<BR>\n" );
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document.write( "\"outer rectangle\" then is:<BR>\n" );
document.write( "(2x+18)(2x+13) <BR>\n" );
document.write( "= 4x^2 + 26x + 36x + 234<BR>\n" );
document.write( "= 4x^2 + 62x + 234<BR>\n" );
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document.write( "\"Inner rectangle\" is:<BR>\n" );
document.write( "18 * 13 = 234<BR>\n" );
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document.write( "Now, instead of:<BR>\n" );
document.write( "\"outer rectangle\" - \"inner rectangle\" = \"area of gravel path\"<BR>\n" );
document.write( "we have:<BR>\n" );
document.write( "(4x^2 + 62x + 234) - 234 = 516<BR>\n" );
document.write( "4x^2 + 62x = 516<BR>\n" );
document.write( "4x^2 + 62x - 516 = 0<BR>\n" );
document.write( "2x^2 + 31x - 258 = 0<BR>\n" );
document.write( "Since, it is difficult to factor, use the quadratic equation.  It will yield:<BR>\n" );
document.write( "x = {6, -21.5}<BR>\n" );
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document.write( "Since the negative solution does not make sense, throw it out leaving:<BR>\n" );
document.write( "x = 6 feet (width of gravel path)<BR>\n" );
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document.write( "Details of quadratic:<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B31x%2B-258+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B31x%2B-258+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%2831%29%5E2-4%2A2%2A-258=3025 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2831%29%5E2-4%2A2%2A-258=3025\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=3025 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-31%2B-sqrt%28+3025+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-31%2B-sqrt%28+3025+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%2831%29%2Bsqrt%28+3025+%29%29%2F2%5C2+=+6 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2831%29%2Bsqrt%28+3025+%29%29%2F2%5C2+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%2831%29-sqrt%28+3025+%29%29%2F2%5C2+=+-21.5 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2831%29-sqrt%28+3025+%29%29%2F2%5C2+=+-21.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B31x%2B-258 ALIGN=MIDDLE  ALT=\"2x%5E2%2B31x%2B-258\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B31x%2B-258+=+2%28x-6%29%2A%28x--21.5%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B31x%2B-258+=+2%28x-6%29%2A%28x--21.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 6, -21.5.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B31%2Ax%2B-258+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B31%2Ax%2B-258+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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