document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=101752>Question 101752</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangle has a diagonal of 15 cm. The width of the rectangle is 3 cm less than its lenght. Find the area of the rectangle. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #73989 by </B><A HREF=/tutors/aboutme.mpl?userid=doukungfoo><B>doukungfoo(195)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=doukungfoo><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=doukungfoo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=73989\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ok first lets sketch out what we know<BR>\n" );
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document.write( "From the drawing we can see that the diagonal forms a right triangle.<BR>\n" );
document.write( "So we can use the pythagorean formula to solve for x.<BR>\n" );
document.write( "The pythagorean formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where c is the hypotenuse of a right triangle. a and b are the other two sides of a right triangle.<BR>\n" );
document.write( "So lets start:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x-3%29%5E2=15%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x-3%29%5E2=15%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "lets start on the right side and go ahead and square 15<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x-3%29%5E2=225 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x-3%29%5E2=225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "now on the left side square (x-3) use the foil method to do this <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2-3x-3x%2B9=225 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2-3x-3x%2B9=225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "combine like terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-6x%2B9=225 ALIGN=MIDDLE  ALT=\"2x%5E2-6x%2B9=225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Ok the pythagorean formula got us this far but now we need to use the quadratic formula to continue solving for x. To do this we first need to set the the equation equal to zero. so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-6x%2B9=225 ALIGN=MIDDLE  ALT=\"2x%5E2-6x%2B9=225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "subtract 225 from both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-6x-216=0 ALIGN=MIDDLE  ALT=\"2x%5E2-6x-216=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "now we can use the quadratic formula to solve for x<BR>\n" );
document.write( "<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=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-6x%2B-216+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-6x%2B-216+=+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=%28-6%29%5E2-4%2A2%2A-216=1764 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-6%29%5E2-4%2A2%2A-216=1764\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1764 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--6%2B-sqrt%28+1764+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--6%2B-sqrt%28+1764+%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-%28-6%29%2Bsqrt%28+1764+%29%29%2F2%5C2+=+12 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+1764+%29%29%2F2%5C2+=+12\"   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-%28-6%29-sqrt%28+1764+%29%29%2F2%5C2+=+-9 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-6%29-sqrt%28+1764+%29%29%2F2%5C2+=+-9\"   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-6x%2B-216 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-6x%2B-216\"   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-6x%2B-216+=+2%28x-12%29%2A%28x--9%29 ALIGN=MIDDLE  ALT=\"2x%5E2%2B-6x%2B-216+=+2%28x-12%29%2A%28x--9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 12, -9.\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-6%2Ax%2B-216+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-6%2Ax%2B-216+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Ok so when solving quadratic equations you get two answers.<BR>\n" );
document.write( "the answers we got here are x = -9 and x = 12<BR>\n" );
document.write( "They are both correct answers mathematically.<BR>\n" );
document.write( "But for our problem we are trying to find the length and width of a rectangle and neither of these can be a negative number.<BR>\n" );
document.write( "So we will throw out the x = -9 answer as an 'extraneous solution' <BR>\n" );
document.write( "That leaves just x = 12<BR>\n" );
document.write( "Ok back to the rectangle from the diagram we started with we can see that the:<BR>\n" );
document.write( "Length = x    and the  width = x - 3<BR>\n" );
document.write( "so<BR>\n" );
document.write( "Length = 12   and the width = 12 - 3 = 9<BR>\n" );
document.write( "Find area of rectangle by Length times width<BR>\n" );
document.write( "12 * 9 = 108<BR>\n" );
document.write( "So finally the area of the rectangle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=108cm%5E2 ALIGN=MIDDLE  ALT=\"108cm%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );