document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=282144>Question 282144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular piece of picture matting is 2 inches longer than it is wide. A rectangular region is cut out of the middle for a picture, leaving 1 inch of matting all around. If the area of the region that is removed is 24 square inches, what are the dimensions of the matting? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #205576 by </B><A HREF=/tutors/aboutme.mpl?userid=Deina><B>Deina(147)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Deina><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=Deina><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=205576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First we will find the dimensions of the piece removed.\r<BR>\n" );
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document.write( "The formula for the area of a rectangle is:<BR>\n" );
document.write( "L * W = 24<BR>\n" );
document.write( "L = W + 2<br><BR>\n" );
document.write( "Substitue the <b>value</b> of L for L<BR>\n" );
document.write( "(W + 2) * W = 24<br><BR>\n" );
document.write( "Expand:<BR>\n" );
document.write( "W^2 + 2W = 24<br><BR>\n" );
document.write( "Subtract 24 from each side:<BR>\n" );
document.write( "W^2 + 2W - 24 = 24 - 24<br><BR>\n" );
document.write( "Solve the quadratic equation:<BR>\n" );
document.write( "W^2 + 2W - 24 = 0<br><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=aw%5E2%2Bbw%2Bc=0 ALIGN=MIDDLE  ALT=\"aw%5E2%2Bbw%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1w%5E2%2B2w%2B-24+=+0 ALIGN=MIDDLE  ALT=\"1w%5E2%2B2w%2B-24+=+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=w%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"w%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=%282%29%5E2-4%2A1%2A-24=100 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%282%29%5E2-4%2A1%2A-24=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=100 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+=+%28-2%2B-sqrt%28+100+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-2%2B-sqrt%28+100+%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=w%5B1%5D+=+%28-%282%29%2Bsqrt%28+100+%29%29%2F2%5C1+=+4 ALIGN=MIDDLE  ALT=\"w%5B1%5D+=+%28-%282%29%2Bsqrt%28+100+%29%29%2F2%5C1+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5B2%5D+=+%28-%282%29-sqrt%28+100+%29%29%2F2%5C1+=+-6 ALIGN=MIDDLE  ALT=\"w%5B2%5D+=+%28-%282%29-sqrt%28+100+%29%29%2F2%5C1+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1w%5E2%2B2w%2B-24 ALIGN=MIDDLE  ALT=\"1w%5E2%2B2w%2B-24\"   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=1w%5E2%2B2w%2B-24+=+1%28w-4%29%2A%28w--6%29 ALIGN=MIDDLE  ALT=\"1w%5E2%2B2w%2B-24+=+1%28w-4%29%2A%28w--6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 4, -6.\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+1%2Ax%5E2%2B2%2Ax%2B-24+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-24+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Since -6 isn't a reasonable answer considering our problem, the <b>width</b> of the <b>inner rectangle</b> is 4\" and the length is 6\".<BR>\n" );
document.write( "W = 4, L = 6<br><BR>\n" );
document.write( "Since there was 1\" margin on <b>all four sides</b> we add 2\" to W and L to get the dimensions of the matting<BR>\n" );
document.write( "W + 2 = 6, L + 2 = 8.<br><BR>\n" );
document.write( "The overall dimensions of the matting are 6\" wide by 8\" long  and bob's your uncle!<BR>\n" );
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