document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=521326>Question 521326</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The area of a rectangle is 270 squared cm. If the shorter side was reduced by 2cm and the longer side was increased by 4cm then the area would increase by 16 squared cm. Find the length of the sides of the original rectangle. (Thank you) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #469158 by </B><A HREF=/tutors/aboutme.mpl?userid=RiiRiiDaDragon><B>RiiRiiDaDragon(1)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=RiiRiiDaDragon><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=RiiRiiDaDragon><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=469158\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hey the previous tutor gave you the correct answers but the steps are totally missing... So in case you're still left in the dark i'll give you my 2 cents\r<BR>\n" );
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document.write( "Okay!<BR>\n" );
document.write( "So lets call the longer side \"x\" and the shorter side \"y\"<BR>\n" );
document.write( "Regardless of which is length or width the formula is the same. \r<BR>\n" );
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document.write( "Area = length x width\r<BR>\n" );
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document.write( "Therefore Area = xy\r<BR>\n" );
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document.write( "The question says that the original area is 270cm^2<BR>\n" );
document.write( "So xy = 270cm^2\r<BR>\n" );
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document.write( "*NB<BR>\n" );
document.write( "Keep in mind, we can manipulate this equation as we have a like depending on what we need so <BR>\n" );
document.write( "x = 270\y<BR>\n" );
document.write( "and<BR>\n" );
document.write( "y = 270\x<BR>\n" );
document.write( "Get it? Got it? Good<BR>\n" );
document.write( "Alright moving on!!\r<BR>\n" );
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document.write( "So the question tells you to increase the longer side (x) by 4cm and to decrease the shorter side (y) by 2 cm and the 'new' area will increase by 16cm^2<BR>\n" );
document.write( "So in a form we can actually use, this basically means:\r<BR>\n" );
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document.write( "(x+4)(y-2)=270+16<BR>\n" );
document.write( "(x+4)(y-2)=286\r<BR>\n" );
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document.write( "Alright so we'll expand this (i'm gonna assume that you know how to expand)and then we'll get<BR>\n" );
document.write( "xy-2x+4y-8=286\r<BR>\n" );
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document.write( "Still with me? Alright kl\r<BR>\n" );
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document.write( "So in order to work this with a quadratic formula we need to change some stuff around<BR>\n" );
document.write( "Remember back when i said that although 270 = xy we can alter it as we see fit to get what we need?<BR>\n" );
document.write( "Yeah .. if you don't i suggest scrolling back up cause this is where it comes in<BR>\n" );
document.write( "(^.^)\r<BR>\n" );
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document.write( "x = 270\y<BR>\n" );
document.write( "and<BR>\n" );
document.write( "y = 270\x\r<BR>\n" );
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document.write( "For this equation i'm gonna write y in terms of x which is this thingamajiggy below:<BR>\n" );
document.write( "y = 270\x<BR>\n" );
document.write( "and inject it into the equation\r<BR>\n" );
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document.write( "Stay with me\r<BR>\n" );
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document.write( "so instead of this:<BR>\n" );
document.write( "xy-2x+4y-8=286\r<BR>\n" );
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document.write( "I get this:<BR>\n" );
document.write( "x(270\x)-2x+4(270\x)-8=286\r<BR>\n" );
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document.write( "We have to simplify that first and get<BR>\n" );
document.write( "270-2x+1080\x)-8=286\r<BR>\n" );
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document.write( "And then put everything as a fraction of x( as in make x the denominator by multiplying everything on the left side of the equation except '1080\" by x<BR>\n" );
document.write( "So we're left with<BR>\n" );
document.write( "(270x-2x^2+1080-8x)\x = 286\r<BR>\n" );
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document.write( "Alright. I always hated fractions... ugh so lets get rid of the x denominator by multiplying both sides by x. This cancels out the x denominator and multiplies the 286 by x on the left side of the equation.\r<BR>\n" );
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document.write( "270x-2x^2+1080-8x= 286x\r<BR>\n" );
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document.write( "Let's switch things up a bit so it looks more like a quadratic equation\r<BR>\n" );
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document.write( "-2x^2+270x-8x+1080=286x\r<BR>\n" );
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document.write( "All quadratic equation need to be equal to zero before we can implement the formula so subtract 286x from both sides\r<BR>\n" );
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document.write( "-2x^2+270x-8x+1080=286x-286x<BR>\n" );
document.write( "-2x^2+270x-8x+1080-286x=0\r<BR>\n" );
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document.write( "That gives us<BR>\n" );
document.write( "-2x^2-24x+1080=0\r<BR>\n" );
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document.write( "So FINALLY  we can use the quadratic formula to solve the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<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=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-24x%2B1080+=+0 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B-24x%2B1080+=+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=%28-24%29%5E2-4%2A-2%2A1080=9216 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-24%29%5E2-4%2A-2%2A1080=9216\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=9216 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+9216+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--24%2B-sqrt%28+9216+%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-%28-24%29%2Bsqrt%28+9216+%29%29%2F2%5C-2+=+-30 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-24%29%2Bsqrt%28+9216+%29%29%2F2%5C-2+=+-30\"   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-%28-24%29-sqrt%28+9216+%29%29%2F2%5C-2+=+18 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-24%29-sqrt%28+9216+%29%29%2F2%5C-2+=+18\"   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-24x%2B1080 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B-24x%2B1080\"   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-24x%2B1080+=+-2%28x--30%29%2A%28x-18%29 ALIGN=MIDDLE  ALT=\"-2x%5E2%2B-24x%2B1080+=+-2%28x--30%29%2A%28x-18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -30, 18.\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-24%2Ax%2B1080+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-2%2Ax%5E2%2B-24%2Ax%2B1080+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Whether you used the quadratic solver i so lovingly offered or did it by hand or calculator or prayer, your answers should be <BR>\n" );
document.write( "x=18 or x= -30\r<BR>\n" );
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document.write( "Distance can't be negative so the -30 is useless to us. <BR>\n" );
document.write( "Now what we focus on is the POSITIVE 18!!\r<BR>\n" );
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document.write( "So with x=18<BR>\n" );
document.write( "We draw for the old equation<BR>\n" );
document.write( " xy=270<BR>\n" );
document.write( "Now that we know what x is its now<BR>\n" );
document.write( "18y=270<BR>\n" );
document.write( "y=270\18<BR>\n" );
document.write( "y=15\r<BR>\n" );
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document.write( "REALLY HOPE THIS HELPED... I KNOW ITS LONG BUT IT SHOULD GET THE JOB DONE <BR>\n" );
document.write( "=) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );