document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=917290>Question 917290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I really need your help in solving this problem. I appreciate your time and efforts. Have a terrific day :) Here is the question?<BR>\n" );
document.write( "Tubing A piece of cardboard in the shape of a parallelogram is twisted to form the tube. The parallelogram has an area of 60 square inches. If its height h, is 7 inches more than the length of the base b, what is the length of the base? (Hint: The Formula for the area of a parallelogram is A = bh.) I Really need your help in solving this!!!!! Thank you so much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #556617 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=556617\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is a parallelogram with a \"base\" measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches and a \"height\" measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=b%2B7 ALIGN=MIDDLE  ALT=\"h=b%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches.<BR>\n" );
document.write( "The area of that parallelogram, in square inches, calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=b%2Ah ALIGN=MIDDLE  ALT=\"A=b%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=b%28b%2B7%29 ALIGN=MIDDLE  ALT=\"A=b%28b%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we know that is 60 square inches, so our equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28b%2B7%29=60 ALIGN=MIDDLE  ALT=\"b%28b%2B7%29=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is, indeed, a quadratic equation.<BR>\n" );
document.write( "It be written in many equivalent forms, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28b%2B7%29=60 ALIGN=MIDDLE  ALT=\"b%28b%2B7%29=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b=60 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b-60=0 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and infinitely many other ways.<BR>\n" );
document.write( " <BR>\n" );
document.write( "There are many ways to solve the quadratic equation.<BR>\n" );
document.write( "One of them if \"by factoring\", which involves knowing factors of 60.<BR>\n" );
document.write( "A fifth grader that realizes that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A12=60 ALIGN=MIDDLE  ALT=\"5%2A12=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "would figure that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=5%29 ALIGN=MIDDLE  ALT=\"highlight%28b=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B7=5%2B7=12 ALIGN=MIDDLE  ALT=\"b%2B7=5%2B7=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution.<BR>\n" );
document.write( "Your teacher, however,<BR>\n" );
document.write( "would expect you to start from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b-60=0 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > form of the equation,<BR>\n" );
document.write( "factor the left hand side into two factors to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B12%29%28b-5%29=0 ALIGN=MIDDLE  ALT=\"%28b%2B12%29%28b-5%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find that the solutions of the equation are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-12 ALIGN=MIDDLE  ALT=\"b=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B12=0 ALIGN=MIDDLE  ALT=\"b%2B12=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > . which makes the first factor zero, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=5%29 ALIGN=MIDDLE  ALT=\"highlight%28b=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b-5=0 ALIGN=MIDDLE  ALT=\"b-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which makes the second factor zero.<BR>\n" );
document.write( "Since the measurement of the base is a positive number, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-12 ALIGN=MIDDLE  ALT=\"b=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes no sense, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=5%29 ALIGN=MIDDLE  ALT=\"highlight%28b=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only sensible answer.<BR>\n" );
document.write( "The length of the base is<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%285inches%29 ALIGN=MIDDLE  ALT=\"highlight%285inches%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "You could also solve the quadratic equation by \"completing the square\" or by using the quadratic formula.<BR>\n" );
document.write( "Completing the square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b=60 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b%2B%287%2F2%29%5E2=60%2B%287%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b%2B%287%2F2%29%5E2=60%2B%287%2F2%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=%28b%2B7%2F2%29%5E2=60%2B49%2F4 ALIGN=MIDDLE  ALT=\"%28b%2B7%2F2%29%5E2=60%2B49%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B7%2F2%29%5E2=240%2F40%2B49%2F4 ALIGN=MIDDLE  ALT=\"%28b%2B7%2F2%29%5E2=240%2F40%2B49%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B7%2F2%29%5E2=289%2F4 ALIGN=MIDDLE  ALT=\"%28b%2B7%2F2%29%5E2=289%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2B7%2F2%29%5E2=%2817%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28b%2B7%2F2%29%5E2=%2817%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So,<BR>\n" );
document.write( "either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B7%2F2=17%2F2 ALIGN=MIDDLE  ALT=\"b%2B7%2F2=17%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=17%2F2-7%2F2 ALIGN=MIDDLE  ALT=\"b=17%2F2-7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=10%2F2 ALIGN=MIDDLE  ALT=\"b=10%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=5%29 ALIGN=MIDDLE  ALT=\"highlight%28b=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2B7%2F2=-17%2F2 ALIGN=MIDDLE  ALT=\"b%2B7%2F2=-17%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-17%2F2-7%2F2 ALIGN=MIDDLE  ALT=\"b=-17%2F2-7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-24%2F2 ALIGN=MIDDLE  ALT=\"b=-24%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-12 ALIGN=MIDDLE  ALT=\"b=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is not a good length for the base of a parallelogram.<BR>\n" );
document.write( "Using the quadratic formula, which says that <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;\" > gives the solutions to <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;\" > .<BR>\n" );
document.write( "Applied to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x-60=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B7x-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=7 ALIGN=MIDDLE  ALT=\"b=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-60 ALIGN=MIDDLE  ALT=\"c=-60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x-60=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B7x-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-7+%2B-+sqrt%287%5E2-4%2A1%2A%28-60%29+%29%29%2F%282%2A1%29=%28-7+%2B-+sqrt%2849%2B240%29%29%2F2=%28-7+%2B-+sqrt%28289%29%29%2F2=%28-7+%2B-+17%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-7+%2B+17%29%2F2=10%2F2=5 ALIGN=MIDDLE  ALT=\"x=%28-7+%2B+17%29%2F2=10%2F2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-7+-+17%29%2F2=%28-24%29%2F2=-12 ALIGN=MIDDLE  ALT=\"x=%28-7+-+17%29%2F2=%28-24%29%2F2=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, since the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x-60=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B7x-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-12 ALIGN=MIDDLE  ALT=\"x=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B7b-60=0 ALIGN=MIDDLE  ALT=\"b%5E2%2B7b-60=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-12 ALIGN=MIDDLE  ALT=\"b=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > . (I just change the name of the variable used for the base so that ift would not get confused with the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <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;\" >and <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;\" > ).<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTES:<BR>\n" );
document.write( "What is the base and what is the height of a parallelogram (or a triangle) is<BR>\n" );
document.write( "\"in the eye of the beholder\". We think of the base as what we stand something on so that it will stay stable, and the height as the perpendicular measurement to the farthest point from the base. However, for calculation purposes, we use as base whatever makes calculations easier.<BR>\n" );
document.write( "Your parallelogram looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-1%2C19%2C-2%2C13%2C%0D%0Aline%280%2C0%2C5%2C0%29%2Cline%2812.04%2C12%2C17.04%2C12%29%2C%0D%0Aline%280%2C0%2C12.04%2C12%29%2Cline%285%2C0%2C17.04%2C12%29%2Clocate%282.5%2C0%2C5%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > so it looks a bit unstable if you stand it on that \"base\". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );