document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=164879>Question 164879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with the following problem.  Consider the function f(x)=x^2+6x-2.  Find h, the x-coordinate of the vertex of this parabola. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #121510 by </B><A HREF=/tutors/aboutme.mpl?userid=KnightOwlTutor><B>KnightOwlTutor(293)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KnightOwlTutor><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=KnightOwlTutor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=121510\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You first have to complete the square. \r<BR>\n" );
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document.write( "0=x^2+6x-2<BR>\n" );
document.write( "Add 2 to both sides\r<BR>\n" );
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document.write( "2=x^2+6x\r<BR>\n" );
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document.write( "Take the 6 and half it. Then take the value in this case it is 3 and square it.add 9 to both sides.\r<BR>\n" );
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document.write( "2+9=(x+3)^2<BR>\n" );
document.write( "0=(x+3)^2-11\r<BR>\n" );
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document.write( "The vertex is at (3,-11)\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B6x%2B-2+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%2B-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
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document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%286%29%5E2-4%2A1%2A-2=44 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%286%29%5E2-4%2A1%2A-2=44\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=44 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-6%2B-sqrt%28+44+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-6%2B-sqrt%28+44+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%286%29%2Bsqrt%28+44+%29%29%2F2%5C1+=+0.3166247903554 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%286%29%2Bsqrt%28+44+%29%29%2F2%5C1+=+0.3166247903554\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%286%29-sqrt%28+44+%29%29%2F2%5C1+=+-6.3166247903554 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%286%29-sqrt%28+44+%29%29%2F2%5C1+=+-6.3166247903554\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B6x%2B-2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%2B-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B6x%2B-2+=+1%28x-0.3166247903554%29%2A%28x--6.3166247903554%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B6x%2B-2+=+1%28x-0.3166247903554%29%2A%28x--6.3166247903554%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.3166247903554, -6.3166247903554.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B6%2Ax%2B-2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B6%2Ax%2B-2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " </SPAN>\n" );
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