document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117704>Question 117704</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the vertex of this quadratic equation?<BR>\n" );
document.write( "y=x^2+6x+8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85696 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=85696\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First, it helps if you can picture it. Does it look like a pointy<BR>\n" );
document.write( "cup that holds water, or is it a pointy cup that poured all its <BR>\n" );
document.write( "water out?<BR>\n" );
document.write( "If the number in front of <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;\" > is positive, it's a pointy cup<BR>\n" );
document.write( "that holds water. In this case, The number in front of <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;\" ><BR>\n" );
document.write( "is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%5E2+=+x%5E2 ALIGN=MIDDLE  ALT=\"1%2Ax%5E2+=+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now you can picture it.<BR>\n" );
document.write( "(1) If it touches the x-axis it has 1 real solution for x.<BR>\n" );
document.write( "That means you can somehow get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the form <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+r%29%5E2+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+r%29%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+r ALIGN=MIDDLE  ALT=\"x+=+r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only solution.<BR>\n" );
document.write( "The other possibilities are:<BR>\n" );
document.write( "(2) It floats above the x-axis without touching it.<BR>\n" );
document.write( "(3) It crosses the x-axis in 2 places<BR>\n" );
document.write( "That's almost all you need to know. Now picture it crossing the x-axis<BR>\n" );
document.write( "in 2 places. Where is the vertex? It's exactly in the middle between<BR>\n" );
document.write( "the 2 crossing points. The crossing points are the roots<BR>\n" );
document.write( "So, if the roots were at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2B1 ALIGN=MIDDLE  ALT=\"x+=+%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2B3 ALIGN=MIDDLE  ALT=\"x+=+%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the vertex<BR>\n" );
document.write( "would be at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2B2 ALIGN=MIDDLE  ALT=\"x+=+%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now all you need to know are the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "They occur at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0 ALIGN=MIDDLE  ALT=\"y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+6x+%2B+8+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+6x+%2B+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Right away I see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2A+4+=+8 ALIGN=MIDDLE  ALT=\"2+%2A+4+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+%2B+4+=+6 ALIGN=MIDDLE  ALT=\"2+%2B+4+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That leads me to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+4%29%28x+%2B+2%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+4%29%28x+%2B+2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What makes this true?<BR>\n" );
document.write( "Either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-4 ALIGN=MIDDLE  ALT=\"x+=+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > These are the roots. What's<BR>\n" );
document.write( "exactly in the middle? <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;\" > is right between them.<BR>\n" );
document.write( "So, that's where the vertex is. A point needs an x and a y, so what<BR>\n" );
document.write( "is the y? Plug <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;\" > into the original equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%5E2+%2B+6x+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+x%5E2+%2B+6x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%29%5E2+%2B+6%2A%28-3%29+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+%28-3%29%5E2+%2B+6%2A%28-3%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+9+-+18+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+9+-+18+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-1 ALIGN=MIDDLE  ALT=\"y+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, the vertex is at (-3,-1)<BR>\n" );
document.write( "Here's the graph<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2+%2B+6x+%2B+8%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2+%2B+6x+%2B+8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );