document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=219646>Question 219646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello I was wondering if you could help me answer this problem.\r<BR>\n" );
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document.write( "         converting to vertex form<BR>\n" );
document.write( "               y=3x^2-36x-117\r<BR>\n" );
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document.write( "thank you for your help :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #165145 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=165145\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-36x-117 ALIGN=MIDDLE  ALT=\"3x%5E2-36x-117\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the right side of the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%5E2-12x-39%29 ALIGN=MIDDLE  ALT=\"3%28x%5E2-12x-39%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This step is very important: the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <font size=4><b>must</b></font> be equal to 1.\r<BR>\n" );
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document.write( "Take half of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coefficient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28-12%29=-6 ALIGN=MIDDLE  ALT=\"%281%2F2%29%28-12%29=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36 ALIGN=MIDDLE  ALT=\"36\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%5E2=%28-6%29%28-6%29=36 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2=%28-6%29%28-6%29=36\"   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=3%28x%5E2-12x%2Bhighlight%2836-36%29-39%29 ALIGN=MIDDLE  ALT=\"3%28x%5E2-12x%2Bhighlight%2836-36%29-39%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now add <font size=4><b>and</b></font> subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36 ALIGN=MIDDLE  ALT=\"36\"   DISABLED_style= \"max-width:90%; height:auto;\" > inside the parenthesis. Make sure to place this after the \"x\" term. Notice how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-36=0 ALIGN=MIDDLE  ALT=\"36-36=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the expression is not changed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28%28x%5E2-12x%2B36%29-36-39%29 ALIGN=MIDDLE  ALT=\"3%28%28x%5E2-12x%2B36%29-36-39%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the first three terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28%28x-6%29%5E2-36-39%29 ALIGN=MIDDLE  ALT=\"3%28%28x-6%29%5E2-36-39%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-12x%2B36 ALIGN=MIDDLE  ALT=\"x%5E2-12x%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-6%29%5E2\"   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=3%28%28x-6%29%5E2-75%29 ALIGN=MIDDLE  ALT=\"3%28%28x-6%29%5E2-75%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-6%29%5E2%2B3%28-75%29 ALIGN=MIDDLE  ALT=\"3%28x-6%29%5E2%2B3%28-75%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So after completing the square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-36x-117 ALIGN=MIDDLE  ALT=\"3x%5E2-36x-117\"   DISABLED_style= \"max-width:90%; height:auto;\" > transforms to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-36x-117=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"3x%5E2-36x-117=3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%5E2-36x-117 ALIGN=MIDDLE  ALT=\"y=3x%5E2-36x-117\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"y=3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"y=3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in vertex form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where (h,k) is the vertex. In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=6 ALIGN=MIDDLE  ALT=\"h=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-225 ALIGN=MIDDLE  ALT=\"k=-225\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the vertex is (6, -225).\r<BR>\n" );
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document.write( "Note: to verify the answer, you could either graph the two equations (and you'll see that they are really the same) or you can expand out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-6%29%5E2-225 ALIGN=MIDDLE  ALT=\"3%28x-6%29%5E2-225\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the original expression again.<BR>\n" );
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