document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=183003>Question 183003</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having trouble finding the vertex of parabolas. I need to find the axes, roots, and the vertices of the 3 following questions. Can you please show it to me in the completing the square method. <BR>\n" );
document.write( "The questions are<BR>\n" );
document.write( "y=2x^2-x+6<BR>\n" );
document.write( "y=x^2-8x+2<BR>\n" );
document.write( "y=4x^2-x+1<BR>\n" );
document.write( "Thanks in advance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #137380 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=137380\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The x-co-ordinate, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bv%5D ALIGN=MIDDLE  ALT=\"x%5Bv%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, of the vertex is exactly midway <BR>\n" );
document.write( "between the roots, so if the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > the vertex is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5B1%5D+%2B+r%5B2%5D%29%2F2 ALIGN=MIDDLE  ALT=\"%28r%5B1%5D+%2B+r%5B2%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D ALIGN=MIDDLE  ALT=\"y%5Bv%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "Another way to find the vertex is, if the equation is in<BR>\n" );
document.write( "the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ax%5E2+%2B+bx+%2B+c ALIGN=MIDDLE  ALT=\"y+=+ax%5E2+%2B+bx+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bv%5D ALIGN=MIDDLE  ALT=\"x%5Bv%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. I'll solve both ways:<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+x+%2B+6 ALIGN=MIDDLE  ALT=\"2x%5E2+-+x+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To complete the square, take 1/2 the co-efficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "square it, and add it to both sides.<BR>\n" );
document.write( "First set equation equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "from both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+x+=+-6 ALIGN=MIDDLE  ALT=\"2x%5E2+-+x+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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+-+%281%2F2%29%2Ax+=+-3 ALIGN=MIDDLE  ALT=\"x%5E2+-+%281%2F2%29%2Ax+=+-3\"   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+-%281%2F2%29%2Ax+%2B+%281%2F4%29%5E2+=+-3+%2B+%281%2F4%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+-%281%2F2%29%2Ax+%2B+%281%2F4%29%5E2+=+-3+%2B+%281%2F4%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+-+%281%2F4%29%29%5E2+=+-%2848%2F16%29+%2B+1%2F16 ALIGN=MIDDLE  ALT=\"%28x+-+%281%2F4%29%29%5E2+=+-%2848%2F16%29+%2B+1%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Take the square root of both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+%281%2F4%29+=+sqrt%2847%29%2F4 ALIGN=MIDDLE  ALT=\"x+-+%281%2F4%29+=+sqrt%2847%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281%2F4%29+%2B-+sqrt%2847%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%281%2F4%29+%2B-+sqrt%2847%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The 2 roots are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+%2B+sqrt%2847%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%281+%2B+sqrt%2847%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%281+-+sqrt%2847%29%29%2F4 ALIGN=MIDDLE  ALT=\"x+=+%281+-+sqrt%2847%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now I find the point midway between the roots<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5B1%5D+%2B+r%5B2%5D%29%2F2+=++%28%28%281+%2B+sqrt%2847%29%29%2F4%29+%2B+%28%281+-+sqrt%2847%29%29%2F4%29%29%2F2 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=%28%28%281+%2B+sqrt%2847%29%29%2F4%29+%2B+%28%281+-+sqrt%2847%29%29%2F4%29%29%2F2%29+=+1%2F8+%2B+1%2F8 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(notice the terms with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2847%29 ALIGN=MIDDLE  ALT=\"sqrt%2847%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancel)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bv%5D+=+1%2F4 ALIGN=MIDDLE  ALT=\"x%5Bv%5D+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And now the easy way:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bv%5D+=+%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x%5Bv%5D+=+%28-b%29%2F%282a%29\"   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+%2B+bx+%2B+c+=+2x%5E2+-+x+%2B+6 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+2x%5E2+-+x+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2 ALIGN=MIDDLE  ALT=\"a+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-1 ALIGN=MIDDLE  ALT=\"b+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bv%5D+=+%28-%28-1%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"x%5Bv%5D+=+%28-%28-1%29%29%2F%282%2A2%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%5Bv%5D+=+1%2F4 ALIGN=MIDDLE  ALT=\"x%5Bv%5D+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now just plug this into the equation to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D ALIGN=MIDDLE  ALT=\"y%5Bv%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++2x%5E2+-+x+%2B+6 ALIGN=MIDDLE  ALT=\"y+=++2x%5E2+-+x+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D+=+2%2A%281%2F4%29%5E2+-+%281%2F4%29+%2B+6 ALIGN=MIDDLE  ALT=\"y%5Bv%5D+=+2%2A%281%2F4%29%5E2+-+%281%2F4%29+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D+=+2%2F16+-+4%2F16+%2B+96%2F16 ALIGN=MIDDLE  ALT=\"y%5Bv%5D+=+2%2F16+-+4%2F16+%2B+96%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D+=+94%2F16 ALIGN=MIDDLE  ALT=\"y%5Bv%5D+=+94%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bv%5D+=+47%2F8 ALIGN=MIDDLE  ALT=\"y%5Bv%5D+=+47%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll plot it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-2%2C+2%2C+-2%2C+7%2C+2x%5E2+-+x+%2B+6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-2%2C+2%2C+-2%2C+7%2C+2x%5E2+-+x+%2B+6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );