document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=286261>Question 286261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a 5 in. by 7 in. photograph is surrounded by a frame of uniform width. The area of the frame equals the area of the photograph. Find the width of the frame </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #207634 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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=207634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A of picture = 5*7=35<BR>\n" );
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document.write( "Frame width = x<BR>\n" );
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document.write( "The area of the frame = area of the picture, which means it is 35+35 = 70<BR>\n" );
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document.write( "(2x+5)(2x+7) = 70<BR>\n" );
document.write( "4x^2 +14x +10x + 35 = 70<BR>\n" );
document.write( "4x^2 +24x + 35 = 70<BR>\n" );
document.write( "4x^2 +24x -35 = 0<BR>\n" );
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document.write( "Using the quadratic solver...<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=4x%5E2%2B24x%2B-35+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B24x%2B-35+=+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=%2824%29%5E2-4%2A4%2A-35=1136 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2824%29%5E2-4%2A4%2A-35=1136\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1136 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+=+%28-24%2B-sqrt%28+1136+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-24%2B-sqrt%28+1136+%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-%2824%29%2Bsqrt%28+1136+%29%29%2F2%5C4+=+1.21307488658818 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2824%29%2Bsqrt%28+1136+%29%29%2F2%5C4+=+1.21307488658818\"   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-%2824%29-sqrt%28+1136+%29%29%2F2%5C4+=+-7.21307488658818 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2824%29-sqrt%28+1136+%29%29%2F2%5C4+=+-7.21307488658818\"   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=4x%5E2%2B24x%2B-35 ALIGN=MIDDLE  ALT=\"4x%5E2%2B24x%2B-35\"   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=4x%5E2%2B24x%2B-35+=+4%28x-1.21307488658818%29%2A%28x--7.21307488658818%29 ALIGN=MIDDLE  ALT=\"4x%5E2%2B24x%2B-35+=+4%28x-1.21307488658818%29%2A%28x--7.21307488658818%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.21307488658818, -7.21307488658818.\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+4%2Ax%5E2%2B24%2Ax%2B-35+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B24%2Ax%2B-35+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "x = 1.2130  (The other root is negative, so it is spurious.)<BR>\n" );
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document.write( "Checking, what is the area of a rectangle defined by the outside of the frame?<BR>\n" );
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document.write( "(5+2(1.2130)) * (7 + 2(1.2130)) = (5+2.426)*(7+2.426) = 7.246*9.246 = 69.997,<BR>\n" );
document.write( "which is pretty close to 70.<BR>\n" );
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document.write( "Done. </SPAN>\n" );
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