document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=40285>Question 40285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of a picture is 1 inch less than twice its width.  The frame around the picture has a uniform width of 2 inches and an area of 96 square inches.  What are the dimensions of the picture? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #25684 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=25684\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The length of a picture is 1 inch less than twice its width.  The frame around the picture has a uniform width of 2 inches and an area of 96 square inches.  What are the dimensions of the picture?\r<BR>\n" );
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document.write( "OUTSIDE FRAME DATA:<BR>\n" );
document.write( "Let the width be \"x\"<BR>\n" );
document.write( "Then the length is \"2x-1\" inches.\r<BR>\n" );
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document.write( "PICTURE DATA:<BR>\n" );
document.write( "width = x-2(2)=x-4 inches<BR>\n" );
document.write( "length = 2x-1-2(2)= 2x-5 inches<BR>\n" );
document.write( "EQUATION:<BR>\n" );
document.write( "Picture area = length * width = 96 sq. in.<BR>\n" );
document.write( "(x-4)(2x-5)=96<BR>\n" );
document.write( "2x^2-13x+20=96<BR>\n" );
document.write( "2x^2-13x-76=0<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%2B-13x%2B-76+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-13x%2B-76+=+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=%28-13%29%5E2-4%2A2%2A-76=777 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-13%29%5E2-4%2A2%2A-76=777\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=777 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--13%2B-sqrt%28+777+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--13%2B-sqrt%28+777+%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-%28-13%29%2Bsqrt%28+777+%29%29%2F2%5C2+=+10.2186799323832 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-13%29%2Bsqrt%28+777+%29%29%2F2%5C2+=+10.2186799323832\"   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-%28-13%29-sqrt%28+777+%29%29%2F2%5C2+=+-3.71867993238318 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-13%29-sqrt%28+777+%29%29%2F2%5C2+=+-3.71867993238318\"   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=2x%5E2%2B-13x%2B-76 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-13x%2B-76\"   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%2B-13x%2B-76+=+2%28x-10.2186799323832%29%2A%28x--3.71867993238318%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-13x%2B-76+=+2%28x-10.2186799323832%29%2A%28x--3.71867993238318%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 10.2186799323832, -3.71867993238318.\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%2B-13%2Ax%2B-76+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-13%2Ax%2B-76+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Only x=10.22 is a usuable answer.<BR>\n" );
document.write( "So, dimensions of the picture are:<BR>\n" );
document.write( "width = x-2=8.22 inches<BR>\n" );
document.write( "length = 2x-5=15.44 inches<BR>\n" );
document.write( "cheers,<BR>\n" );
document.write( "Stan H. <BR>\n" );
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