document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=215000>Question 215000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the vertex of the function by completing the square. <BR>\n" );
document.write( "-18x^2 + -177x + -190 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "An explanation of how to go about completing the square would be very much appreciated! Thanks!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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 #162462 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=162462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18x%5E2-177x-190 ALIGN=MIDDLE  ALT=\"-18x%5E2-177x-190\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%5E2%2B%2859%2F6%29x%2B95%2F9%29 ALIGN=MIDDLE  ALT=\"-18%28x%5E2%2B%2859%2F6%29x%2B95%2F9%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=-18 ALIGN=MIDDLE  ALT=\"-18\"   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" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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=59%2F6 ALIGN=MIDDLE  ALT=\"59%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=59%2F12 ALIGN=MIDDLE  ALT=\"59%2F12\"   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%2859%2F6%29=59%2F12 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2859%2F6%29=59%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=59%2F12 ALIGN=MIDDLE  ALT=\"59%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3481%2F144 ALIGN=MIDDLE  ALT=\"3481%2F144\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2859%2F12%29%5E2=%2859%2F12%29%2859%2F12%29=3481%2F144 ALIGN=MIDDLE  ALT=\"%2859%2F12%29%5E2=%2859%2F12%29%2859%2F12%29=3481%2F144\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%5E2%2B%2859%2F6%29x%2Bhighlight%283481%2F144-3481%2F144%29%2B95%2F9%29 ALIGN=MIDDLE  ALT=\"-18%28x%5E2%2B%2859%2F6%29x%2Bhighlight%283481%2F144-3481%2F144%29%2B95%2F9%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=3481%2F144 ALIGN=MIDDLE  ALT=\"3481%2F144\"   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=3481%2F144-3481%2F144=0 ALIGN=MIDDLE  ALT=\"3481%2F144-3481%2F144=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the expression is not changed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28%28x%5E2%2B%2859%2F6%29x%2B3481%2F144%29-3481%2F144%2B95%2F9%29 ALIGN=MIDDLE  ALT=\"-18%28%28x%5E2%2B%2859%2F6%29x%2B3481%2F144%29-3481%2F144%2B95%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group the first three terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28%28x%2B59%2F12%29%5E2-3481%2F144%2B95%2F9%29 ALIGN=MIDDLE  ALT=\"-18%28%28x%2B59%2F12%29%5E2-3481%2F144%2B95%2F9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%2859%2F6%29x%2B3481%2F144 ALIGN=MIDDLE  ALT=\"x%5E2%2B%2859%2F6%29x%2B3481%2F144\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B59%2F12%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B59%2F12%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28%28x%2B59%2F12%29%5E2-1961%2F144%29 ALIGN=MIDDLE  ALT=\"-18%28%28x%2B59%2F12%29%5E2-1961%2F144%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%2B59%2F12%29%5E2-18%28-1961%2F144%29 ALIGN=MIDDLE  ALT=\"-18%28x%2B59%2F12%29%5E2-18%28-1961%2F144%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So after completing the square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18x%5E2-177x-190 ALIGN=MIDDLE  ALT=\"-18x%5E2-177x-190\"   DISABLED_style= \"max-width:90%; height:auto;\" > transforms to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18x%5E2-177x-190=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"-18x%5E2-177x-190=-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18x%5E2-177x-190=0 ALIGN=MIDDLE  ALT=\"-18x%5E2-177x-190=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18%28x%2B59%2F12%29%5E2%2B1961%2F8=0 ALIGN=MIDDLE  ALT=\"-18%28x%2B59%2F12%29%5E2%2B1961%2F8=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-18x%5E2-177x-190 ALIGN=MIDDLE  ALT=\"y=-18x%5E2-177x-190\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is now 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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-18 ALIGN=MIDDLE  ALT=\"a=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-59%2F12 ALIGN=MIDDLE  ALT=\"h=-59%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1961%2F8 ALIGN=MIDDLE  ALT=\"k=1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Remember, the vertex of <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;\" > is (h,k).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the vertex of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8 ALIGN=MIDDLE  ALT=\"y=-18%28x%2B59%2F12%29%5E2%2B1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-\frac{59}{12},\frac{1961}{8}\right)\"> since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-59%2F12 ALIGN=MIDDLE  ALT=\"h=-59%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=1961%2F8 ALIGN=MIDDLE  ALT=\"k=1961%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );