document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=693410>Question 693410</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Finding finding the roots of a quadratic equation with leading coefficient greater than 1\r<BR>\n" );
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document.write( "Solve for u<BR>\n" );
document.write( "3u^2-5u=2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427383 by </B><A HREF=/tutors/aboutme.mpl?userid=pmatei><B>pmatei(79)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=pmatei><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427383\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For quadratics you need everything on the left side of the equal sign:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%5E2-5u-2=0 ALIGN=MIDDLE  ALT=\"3u%5E2-5u-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There are 2 ways of solving quadratics. \r<BR>\n" );
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document.write( "A. You try to find to numbers that their product is equal to the product of the u^2 coefficient and the number without letter (in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-2%29=-6 ALIGN=MIDDLE  ALT=\"3%28-2%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >), and their sum is equal to the u coefficient (in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >). The numbers that multiplied give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and added give <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > are : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2B1 ALIGN=MIDDLE  ALT=\"%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5u ALIGN=MIDDLE  ALT=\"-5u\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the original equation write the two numbers multiplied by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%5E2+-6u%2Bu-2=0 ALIGN=MIDDLE  ALT=\"3u%5E2+-6u%2Bu-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Group the four terms in 2 pairs: first two terms and last two terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283u%5E2-6u%29%2B%28u-2%29=0 ALIGN=MIDDLE  ALT=\"%283u%5E2-6u%29%2B%28u-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "See if you can factor out anything from those parentheses (you always have at least the letter from the first parenthesis):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%28u-2%29%2B1%28u-2%29=0 ALIGN=MIDDLE  ALT=\"3u%28u-2%29%2B1%28u-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you done everything right the two parentheses are the same at this point. This repeated parenthesis is one of the 2 parentheses you want to solve the quadratic. The second one is put together by what you factored out:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28u-2%29%283u%2B1%29=0 ALIGN=MIDDLE  ALT=\"%28u-2%29%283u%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the quadratic solutions take each parenthesis and set it equal to zero.\r<BR>\n" );
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document.write( "First solution: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u-2=0 ALIGN=MIDDLE  ALT=\"u-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=2 ALIGN=MIDDLE  ALT=\"u=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Second solution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%2B1=0 ALIGN=MIDDLE  ALT=\"3u%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=-1%2F3 ALIGN=MIDDLE  ALT=\"u=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This method does not always work. \r<BR>\n" );
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document.write( "B. The second method is to know the quadratic formula, and works every time:\r<BR>\n" );
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document.write( "If <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;\" > then the solutions are given by:\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( "In your case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3u%5E2-5u-2=0 ALIGN=MIDDLE  ALT=\"3u%5E2-5u-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-2 ALIGN=MIDDLE  ALT=\"c=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%285+%2B-+sqrt%28+25-4%2A%283%29%2A%28-2%29+%29%29%2F6+ ALIGN=MIDDLE  ALT=\"u=%285+%2B-+sqrt%28+25-4%2A%283%29%2A%28-2%29+%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%285+%2B-+sqrt%28+25%2B24%29%29%2F6+ ALIGN=MIDDLE  ALT=\"u=%285+%2B-+sqrt%28+25%2B24%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%285+%2B-+sqrt%28+49%29%29%2F6+ ALIGN=MIDDLE  ALT=\"u=%285+%2B-+sqrt%28+49%29%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%285+%2B-+7%29%2F6+ ALIGN=MIDDLE  ALT=\"u=%285+%2B-+7%29%2F6+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First solution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%285%2B7%29%2F6+=+12%2F6=2 ALIGN=MIDDLE  ALT=\"u+=+%285%2B7%29%2F6+=+12%2F6=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Second solution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=%285-7%29%2F6=-2%2F6=-1%2F3 ALIGN=MIDDLE  ALT=\"u=%285-7%29%2F6=-2%2F6=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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