document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083261>Question 1083261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I just learned how to solve quadratic equations by completing the square and my book throws this doozy in a section called \"Challenge\", can this be solved by completing the square? Are there more than one way to solve it?\r<BR>\n" );
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document.write( "Solve for x in terms of a.\r<BR>\n" );
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document.write( "3x^2+ax^2=9x+9a </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697273 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=697273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Some (not all) quadratic equation can be solved by factoring.<BR>\n" );
document.write( "Every quadratic equation can be solved by completing the square,<BR>\n" );
document.write( "and every quadratic equation can be solved by using the quadratic formula.<BR>\n" );
document.write( "People usually resort to the quadratic formula,<BR>\n" );
document.write( "but professor Peter Alfeld of the university of Utah<BR>\n" );
document.write( "believes we should complete squares.<BR>\n" );
document.write( "I use what comes easier. When factoring works, it may be easier.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2Bax%5E2=9x%2B9a ALIGN=MIDDLE  ALT=\"3x%5E2%2Bax%5E2=9x%2B9a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Ba%29x%5E2-9x=9a ALIGN=MIDDLE  ALT=\"%283%2Ba%29x%5E2-9x=9a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Maybe dividing everything by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Ba%29 ALIGN=MIDDLE  ALT=\"%283%2Ba%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at this point would be easier.<BR>\n" );
document.write( "Of course, we will have to add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E-3 ALIGN=MIDDLE  ALT=\"a%3C%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a requirement to any solution we find as a function of a.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%289%2F%283%2Ba%29%29x=9a%2F%283%2Ba%29 ALIGN=MIDDLE  ALT=\"x%5E2-%289%2F%283%2Ba%29%29x=9a%2F%283%2Ba%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-%289%2F%283%2Ba%29%29x%2B%289%2F%282%283%2Ba%29%29%29%5E2=9a%2F%283%2Ba%29%2B%289%2F%282%283%2Ba%29%29%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-9%2F%282a%2B6%29%29%5E2=9a%2F%283%2Ba%29%2B%289%2F%282%283%2Ba%29%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F%282a%2B6%29%29%5E2=9a%2F%283%2Ba%29%2B%289%2F%282%283%2Ba%29%29%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=%28x-9%2F%282a%2B6%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F%282a%2B6%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9a%284%28a%2B3%29%29%2F%282a%2B6%29%5E2 ALIGN=MIDDLE  ALT=\"9a%284%28a%2B3%29%29%2F%282a%2B6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%5E2%2F%282a%2B6%29%5E2 ALIGN=MIDDLE  ALT=\"9%5E2%2F%282a%2B6%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=%28x-9%2F%282a%2B6%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F%282a%2B6%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289%284a%5E2%2B12%29%2B81%29%2F%282a%2B6%29%5E2 ALIGN=MIDDLE  ALT=\"%289%284a%5E2%2B12%29%2B81%29%2F%282a%2B6%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=%28x-9%2F%282a%2B6%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F%282a%2B6%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%284a%5E2%2B12a%2B9%29%2F%282a%2B6%29%5E2 ALIGN=MIDDLE  ALT=\"9%284a%5E2%2B12a%2B9%29%2F%282a%2B6%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=%28x-9%2F%282a%2B6%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-9%2F%282a%2B6%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%282a%2B3%29%5E2%2F%282a%2B6%29%5E2 ALIGN=MIDDLE  ALT=\"9%282a%2B3%29%5E2%2F%282a%2B6%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=x=9%2F%282a%2B6%29+%2B-+3%282a%2B3%29%2F%282a%2B6%29 ALIGN=MIDDLE  ALT=\"x=9%2F%282a%2B6%29+%2B-+3%282a%2B3%29%2F%282a%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%289+%2B-+%286a%2B9%29%29%2F%282a%2B6%29 ALIGN=MIDDLE  ALT=\"x=%289+%2B-+%286a%2B9%29%29%2F%282a%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the two solutions are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=%286a%2B18%29%2F%282a%2B6%29=3 ALIGN=MIDDLE  ALT=\"x%5B1%5D=%286a%2B18%29%2F%282a%2B6%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=6a%2F%282a%2B6%29=3a%2F%28a%2B3%29 ALIGN=MIDDLE  ALT=\"x%5B2%5D=6a%2F%282a%2B6%29=3a%2F%28a%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E-3 ALIGN=MIDDLE  ALT=\"a%3C%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "WAIT!<BR>\n" );
document.write( "Why did we not realize that we could solve by factoring?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2Bax%5E2=9x%2B9a ALIGN=MIDDLE  ALT=\"3x%5E2%2Bax%5E2=9x%2B9a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2-9a%2B3x%5E2-9x=0 ALIGN=MIDDLE  ALT=\"ax%5E2-9a%2B3x%5E2-9x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%28x-3%29=a%289-x%5E2%29 ALIGN=MIDDLE  ALT=\"3x%28x-3%29=a%289-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%5E2-9%29%2B3x%28x-3%29=0 ALIGN=MIDDLE  ALT=\"a%28x%5E2-9%29%2B3x%28x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x%2B3%29%28x-3%29%2B3x%28x-3%29=0 ALIGN=MIDDLE  ALT=\"a%28x%2B3%29%28x-3%29%2B3x%28x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%28x%2B3%29%2B3x%29%28x-3%29=0 ALIGN=MIDDLE  ALT=\"%28a%28x%2B3%29%2B3x%29%28x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ax%2B3a%2B3x%29%28x-3%29=0 ALIGN=MIDDLE  ALT=\"%28ax%2B3a%2B3x%29%28x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2B3%29x-3a%29%28x-3%29=0 ALIGN=MIDDLE  ALT=\"%28%28a%2B3%29x-3a%29%28x-3%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point, either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B3%29x-3a=0 ALIGN=MIDDLE  ALT=\"%28a%2B3%29x-3a=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3a%2F%28a%2B3%29 ALIGN=MIDDLE  ALT=\"x=3a%2F%28a%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as long as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E-3 ALIGN=MIDDLE  ALT=\"a%3C%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );