document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=900370>Question 900370</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three parts and uses the quadratic equation: x^2 - 18x + 72 = 0\r<BR>\n" );
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document.write( "Part a. <BR>\n" );
document.write( "1. Graph the quadratic equation and: <BR>\n" );
document.write( "a. Label and include the ordered pair for the vertex on the graph. <BR>\n" );
document.write( "b. Label and include the ordered pair(s) for the x--intercepts on the graph. <BR>\n" );
document.write( "c. How are the solutions identified from the graph?\r<BR>\n" );
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document.write( "Part b. <BR>\n" );
document.write( "1. Solve the quadratic equation by using the \"completing the square\" method.  <BR>\n" );
document.write( "2. What do you notice about the answer you found here and where the x-intercepts of the graph are in part a?\r<BR>\n" );
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document.write( "Part c. <BR>\n" );
document.write( "1. Solve the quadratic equation by using the quadratic formula. Show all your work. <BR>\n" );
document.write( "2. Compare your answer to what you found in part a and b. What do you notice about the answers?\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #548719 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=548719\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <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=complete_square.solver>COMPLETING THE SQUARE solver for quadratics</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Read <A  rel=nofollow HREF=Fool-Proof-Completing-the-Square-Part-I.lesson>this lesson on completing the square by prince_abubu</A>, if you do not know how to complete the square.<BR>Let's convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-18x%2B72=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-18x%2B72=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to standard form by dividing both sides by 1: <BR>\n" );
document.write( "We have: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-18x%2B72=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-18x%2B72=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \n" );
document.write( "What we want to do now is to change this equation to a complete square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >. How can we find out values of <B>somenumber</B> and <B>othernumber</B> that would make it work?<BR>\n" );
document.write( "Look at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the coefficient in our equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2Bhighlight_red%28+-18%29+%2A+x%2B72=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight_red%28+-18%29+%2A+x%2B72=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of x is -18, we know that -18=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+-18%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+-18%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, we know that our equation can be rewritten as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-18%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-18%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we do not yet know the other number.<BR>\n" );
document.write( "We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-18%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-18%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to our original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-18x%2Bhighlight_green%28+72+%29=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-18x%2Bhighlight_green%28+72+%29=0\"   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=%28x%2B-18%2F2%29%5E2+%2B+othernumber+=+x%5E2%2B-18x%2Bhighlight_red%28+-18%5E2%2F4%2Bothernumber+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  The highlighted <FONT COLOR=red>red part</FONT> must be equal to 72 (highlighted <FONT COLOR=green>green part</FONT>).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%5E2%2F4+%2B+othernumber+=+72 ALIGN=MIDDLE  ALT=\"-18%5E2%2F4+%2B+othernumber+=+72\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+72--18%5E2%2F4+=+-9 ALIGN=MIDDLE  ALT=\"othernumber+=+72--18%5E2%2F4+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, the equation converts to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-18%2F2%29%5E2+%2B+-9+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B-18%2F2%29%5E2+%2B+-9+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-18%2F2%29%5E2+=+9 ALIGN=MIDDLE  ALT=\"%28x%2B-18%2F2%29%5E2+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Our equation converted to a square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-18%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B-18%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (9).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Since the right part 9 is greater than zero, there are two solutions:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-18%2F2%29+=+%2Bsqrt%28+9+%29%2C+%28x%2B-18%2F2%29+=+-sqrt%28+9+%29+%29 ALIGN=MIDDLE  ALT=\"system%28+%28x%2B-18%2F2%29+=+%2Bsqrt%28+9+%29%2C+%28x%2B-18%2F2%29+=+-sqrt%28+9+%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    , or<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-18%2F2%29+=+3%2C+%28x%2B-18%2F2%29+=+-3+%29 ALIGN=MIDDLE  ALT=\"system%28+%28x%2B-18%2F2%29+=+3%2C+%28x%2B-18%2F2%29+=+-3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%2B-18%2F2+=+3%2C+x%2B-18%2F2+=+-3+%29 ALIGN=MIDDLE  ALT=\"system%28+x%2B-18%2F2+=+3%2C+x%2B-18%2F2+=+-3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x+=+3--18%2F2%2C+x+=+-3--18%2F2+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+3--18%2F2%2C+x+=+-3--18%2F2+%29\"   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=system%28+x+=+12%2C+x+=+6+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+12%2C+x+=+6+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: x=12, 6.\n" );
document.write( "</TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " <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_formula.solver>Quadratic Formula</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Let's use the quadratic formula to solve for x:<BR><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Starting with the general quadratic<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <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( "  <BR>\n" );
document.write( "  the general solution using the quadratic equation is:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-18%2Ax%2B72=0 ALIGN=MIDDLE  ALT=\"x%5E2-18%2Ax%2B72=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <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=-18 ALIGN=MIDDLE  ALT=\"b=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=72 ALIGN=MIDDLE  ALT=\"c=72\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28--18+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28--18+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=-18, and c=72<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+sqrt%28+%28-18%29%5E2-4%2A1%2A72+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -18 to get 18<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+sqrt%28+324-4%2A1%2A72+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+sqrt%28+324-4%2A1%2A72+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -18 to get 324  (note: remember when you square -18, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-18%29%5E2=-18%2A-18=324 ALIGN=MIDDLE  ALT=\"%28-18%29%5E2=-18%2A-18=324\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+sqrt%28+324%2B-288+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+sqrt%28+324%2B-288+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A72%2A1 ALIGN=MIDDLE  ALT=\"-4%2A72%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-288 ALIGN=MIDDLE  ALT=\"-288\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+sqrt%28+36+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+sqrt%28+36+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+6%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+6%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B-+6%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B-+6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So now the expression breaks down into two parts<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+%2B+6%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2818+%2B+6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2818+-+6%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2818+-+6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Lets look at the first part:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2818+%2B+6%29%2F2 ALIGN=MIDDLE  ALT=\"x=%2818+%2B+6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=24%2F2 ALIGN=MIDDLE  ALT=\"x=24%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <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;\" > Divide<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So one answer is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <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( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now lets look at the second part:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2818+-+6%29%2F2 ALIGN=MIDDLE  ALT=\"x=%2818+-+6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=12%2F2 ALIGN=MIDDLE  ALT=\"x=12%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So another answer is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So our solutions are:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );