document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982262>Question 982262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor and quadratic formulas.<BR>\n" );
document.write( "I must find the exact solution. If the solutions are irrational then I must find the approximate solution.\r<BR>\n" );
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document.write( "PROBLEM:The width of a rectangle gate is 2 meters (m) larger than the height.the diagonal brace measures the root of 6m. Find the width and height. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #603105 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=603105\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H=Height ALIGN=MIDDLE  ALT=\"H=Height\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=Width=H%2B2m ALIGN=MIDDLE  ALT=\"W=Width=H%2B2m\"   DISABLED_style= \"max-width:90%; height:auto;\" >; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=diagonal ALIGN=MIDDLE  ALT=\"D=diagonal\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29m ALIGN=MIDDLE  ALT=\"sqrt%286%29m\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The height and width are two legs of a right triangle with diagonal as hypotenuse.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E2%2BW%5E2=D%5E2 ALIGN=MIDDLE  ALT=\"H%5E2%2BW%5E2=D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute for W.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E2%2B%28H%2B2m%29%5E2=%28sqrt%286%29m%29%5E2 ALIGN=MIDDLE  ALT=\"H%5E2%2B%28H%2B2m%29%5E2=%28sqrt%286%29m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5E2%2BH%5E2%2B4Hm%2B4m%5E2=6m%5E2 ALIGN=MIDDLE  ALT=\"H%5E2%2BH%5E2%2B4Hm%2B4m%5E2=6m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2H%5E2%2B4Hm-2m%5E2=0 ALIGN=MIDDLE  ALT=\"2H%5E2%2B4Hm-2m%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  BORDER=1>\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=aH%5E2%2BbH%2Bc=0 ALIGN=MIDDLE  ALT=\"aH%5E2%2BbH%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2H%5E2%2B4H%2B-2+=+0 ALIGN=MIDDLE  ALT=\"2H%5E2%2B4H%2B-2+=+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=H%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"H%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=%284%29%5E2-4%2A2%2A-2=32 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%284%29%5E2-4%2A2%2A-2=32\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=32 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+=+%28-4%2B-sqrt%28+32+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-4%2B-sqrt%28+32+%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=H%5B1%5D+=+%28-%284%29%2Bsqrt%28+32+%29%29%2F2%5C2+=+0.414213562373095 ALIGN=MIDDLE  ALT=\"H%5B1%5D+=+%28-%284%29%2Bsqrt%28+32+%29%29%2F2%5C2+=+0.414213562373095\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%5B2%5D+=+%28-%284%29-sqrt%28+32+%29%29%2F2%5C2+=+-2.41421356237309 ALIGN=MIDDLE  ALT=\"H%5B2%5D+=+%28-%284%29-sqrt%28+32+%29%29%2F2%5C2+=+-2.41421356237309\"   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=2H%5E2%2B4H%2B-2 ALIGN=MIDDLE  ALT=\"2H%5E2%2B4H%2B-2\"   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=2H%5E2%2B4H%2B-2+=+2%28H-0.414213562373095%29%2A%28H--2.41421356237309%29 ALIGN=MIDDLE  ALT=\"2H%5E2%2B4H%2B-2+=+2%28H-0.414213562373095%29%2A%28H--2.41421356237309%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.414213562373095, -2.41421356237309.\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%2B4%2Ax%2B-2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B4%2Ax%2B-2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "H=0.4142 meters (approx)<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );