document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=130120>Question 130120</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ok so my question is how can you still use the \"answer\" you get from the Quadratic formula to find the vertex? And this is what i did... \r<BR>\n" );
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document.write( "Problem: y= x^2-4x+6<BR>\n" );
document.write( "i used the quadratic formula...<BR>\n" );
document.write( "x= 4(+or-) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28+4%5E2-4%281%29%286%29+%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28+4%5E2-4%281%29%286%29+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > / 2(1)\r<BR>\n" );
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document.write( "and got  x= 2 - 1.415i  and   x= 2 + 1.415i \r<BR>\n" );
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document.write( "how can i use those conjugate pairs to find the vertex? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #95050 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=95050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the x-coordinate of the vertex, simply average the two roots.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+-+1.415i%2B+2+%2B+1.415i ALIGN=MIDDLE  ALT=\"2+-+1.415i%2B+2+%2B+1.415i\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the two roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2B2%29+%2B%281.415i-+1.415i%29 ALIGN=MIDDLE  ALT=\"%282%2B2%29+%2B%281.415i-+1.415i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+-+0i ALIGN=MIDDLE  ALT=\"4+-+0i\"   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=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "So it turns out that when you add any two complex conjugates together you will get a real answer.\r<BR>\n" );
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document.write( "Now divide the answer by 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F2=2 ALIGN=MIDDLE  ALT=\"4%2F2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of vertex is 2\r<BR>\n" );
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document.write( "note:\r<BR>\n" );
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document.write( "Remember the quadratic formula is \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%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the formula for the x-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice if you add the two roots the square root part cancels out to to give you <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this shows how the quadratic formula and the vertex are related. </SPAN>\n" );
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