document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=941039>Question 941039</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A banner is in the shape of a right triangle of area 63 in^2. The height of the banner is 4 inches less than twice the width of the banner. Find the height and the width of the banner.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #573654 by </B><A HREF=/tutors/aboutme.mpl?userid=macston><B>macston(5194)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=macston><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=macston><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=573654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> W=width of banner; H=height of banner=2W-4; A=Area of banner=1/2WH=63 sq in<BR>\n" );
document.write( "Substitute for H  first equation)in the area equation and solve for W:<BR>\n" );
document.write( "1/2WH=63 sq in<BR>\n" );
document.write( "1/2W(2W-4)=63 sq in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%282W%5E2-4W%29=63+in%5E2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%282W%5E2-4W%29=63+in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%5E2-2W=63+in%5E2 ALIGN=MIDDLE  ALT=\"W%5E2-2W=63+in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 63in^2 from each side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%5E2-2W-63in%5E2=63in%5E2-63in%5E2 ALIGN=MIDDLE  ALT=\"W%5E2-2W-63in%5E2=63in%5E2-63in%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(((W^2-2W-63in^2=0}}}<TABLE  BORDER=1>\n" );
document.write( "\n" );
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%2B-2W%2B-63+=+0 ALIGN=MIDDLE  ALT=\"1W%5E2%2B-2W%2B-63+=+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=%28-2%29%5E2-4%2A1%2A-63=256 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-63=256\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=256 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+256+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--2%2B-sqrt%28+256+%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-%28-2%29%2Bsqrt%28+256+%29%29%2F2%5C1+=+9 ALIGN=MIDDLE  ALT=\"W%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+256+%29%29%2F2%5C1+=+9\"   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-%28-2%29-sqrt%28+256+%29%29%2F2%5C1+=+-7 ALIGN=MIDDLE  ALT=\"W%5B2%5D+=+%28-%28-2%29-sqrt%28+256+%29%29%2F2%5C1+=+-7\"   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%2B-2W%2B-63 ALIGN=MIDDLE  ALT=\"1W%5E2%2B-2W%2B-63\"   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%2B-2W%2B-63+=+1%28W-9%29%2A%28W--7%29 ALIGN=MIDDLE  ALT=\"1W%5E2%2B-2W%2B-63+=+1%28W-9%29%2A%28W--7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 9, -7.\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%2B-2%2Ax%2B-63+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-63+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "So W=9,-7<BR>\n" );
document.write( "First Let's consider 9<BR>\n" );
document.write( "When W=9, H=2W-4=2(9)-4=14<BR>\n" );
document.write( "ANSWER Width is 9 in and Height is 14 in, a tall narrow right triangle pointing upward and to the right.<BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "Area=1/2WH=63 in sq<BR>\n" );
document.write( "Area=1/2(9in)(14in)=63 in sq<BR>\n" );
document.write( "Area=1/2(126in sq)=63 in sq<BR>\n" );
document.write( "Area=63 in sq=63 in sq, Check<BR>\n" );
document.write( "Next, let's consider -7.  At first this seems an unlikely width, but if we consider on an X,Y co-ordinate plane right is the positive direction, it means this banner points left. So solving for height:<BR>\n" );
document.write( "H=2W-4=2(-7)-4=-14-4=-18, again negative, so pointing down<BR>\n" );
document.write( "So we have a banner of Width=7 in and Height=18 in pointing left and downward<BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "Area=1/2WH=1/2(-7in)(-18 in)=63 in sq<BR>\n" );
document.write( "Area=1/2(126 in sq)=63in sq<BR>\n" );
document.write( "Area= 63 in sq=63 in sq Check </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );